THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

DAVIS 


GIFT  OF 

PROFESSOR  H.B.  WALKER 


PHYSICAL 
MEASUREMENTS 


DUFF  and  EWELL 


BLAKISTON'S  SCIENCE   SERIES 

PHYSICAL 
MEASUREMENTS 


BY 

A.  WILMER  DUFF 

PROFESSOR  OF   PHYSICS   IN  THE  WORCESTER   POLYTECHNIC   INSTITUTE 


AND 

ARTHUR  W.   EWELL 

PROFESSOR   OF   PHYSICS   IN   THE   WORCESTER  POLYTECHNIC   INSTITUTE 


SECOND  EDITION,  REVISED  AND  ENLARGED 
WITH  78  ILLUSTRATIONS 


PHILADELPHIA 

P.  BLAKISTON'S  SON  &  CO 

1012  WALNUT  STREET 
1910 


LIBRARY 

(JNfVFRSITY  nr  rmtr/\nutn 


COPYRIGHT,  1910,  BY  P.  BLAKISTON'S  SON  &  Co. 


Printed   by 

The  Maple  Press 

York,  Pa. 


PREFACE. 


Our  intention  in  writing  this  book  was  not  to  give  an 
account  of  physical  laboratory  methods  in  general,  but  to 
describe  a  limited  number  of  carefully  chosen  exercises  such 
as  we  have  found  in  our  experience  to  be  suitable  for  the 
laboratory  work  of  students  who  have  had  a  fair  course 
in  General  College  Physics. 

The  descriptions  of  the  exercises  will  usually  fit  apparatus 
and  conditions  of  considerable  diversity,  but  many  practical 
details  have  been  included  where  experience  has  shown  that 
they  are  necessary.  Other  instructors  who  may  adopt  the 
book  will  probably  find  some  of  the  exercises  unsuited  to 
their  classes,  but  the  list  is  sufficiently  extensive  to  afford 
a  considerable  variety  of  selection. 

The  descriptions  of  apparatus  are  intended  to  be  read  by 
the  student  with  the  apparatus  before  him.  Hence  elaborate 
illustrations  have  been  thought  unnecessary.  For  an  ex- 
tended account  of  certain  special  topics,  such  as  the  theory 
of  the  balance  and  the  construction  of  galvanometers, 
references  to  other  works  have  been  given. 

Usually  several  text-books  and  special  treatises  have 
been  referred  to  at  the  beginning  of  the  account  of  an  experi- 
ment. It  is  assumed  that  each  student  will  have  one  of  the 
text-books  and  that  some  of  the  special  works  will  be  found  in 
the  reference  room  of  the  laboratory.  While  the  reference 
is  generally  to  the  latest  edition  (at  the  present  date,  1910), 
those  who  have  different  editions  will  have  no  difficulty  in 
finding  the  passages  referred  to.  Each  instructor  who  uses 

v 


VI  PREFACE 

the  book  will  exercise  his  discretion  as  to  what  preliminary 
reading  will  be  required  and  will  issue  the  necessary  instruc- 
tions to  his  class. 

We  are  indebted  to  Dr.  Albert  W.  Hull  for  assistance  in 
reading  the  page  proof.  Many  of  the  tables  have  been  taken 
from  Ewell's  Physical  Chemistry. 


CONTENTS. 


PAGE 

GENERAL  INTRODUCTION i 

i.  Purpose  of  Course.  2.  General  Directions.  3.  Reports. 
4.  Errors.  5.  Errors  of  Observation.  6.  Possible  Error 
of  a  Calculated  Result.  7.  General  Method  for  the  Possible 
Error  of  a  Result.  8.  Some  General  Notes  on  Errors.  9. 
Probable  Error  of  a  Mean.  10.  Limits  to  Calculations,  n. 
Notation  of  Large  and  Small  quantities.  12.  Plotting  of 
Curves. 

MECHANICS 13 

13.  The  Use  of  a  Vernier.  14.  Vernier  Caliper.  15.  Microm- 
eter Caliper.  16.  Micrometer  Microscope.  17.  Com- 
parator. 1 8.  Spherometer.  19.  Dividing  Engine.  20. 
Cathetometer.  21.  Barometer.  22.  The  Balance.  23.  Ad- 
justment of  Telescope  and  Scale.  24.  Time  Determination. 

I.  To  Make  and  Calibrate  a  Scale. 
II.  Errors  of  Weights. 

III.  Volume,  Mass,  and  Density  of  a  Regular  Solid. 

IV.  Mohr-Westphal  Specific  Gravity  Balance. 
V.  Density  by  the  Volumenometer. 

VI.  Density  of  Gases. 

VII.  Acceleration  of  Gravity  by  Pendulum. 

VIII.  Coefficient  of  Friction. 

IX.  Hooke's  Law  and  Young's  Modulus. 

X.  Rigidity  (or  Shear  Modulus). 

XL  Viscosity. 

XII.  Surface  Tension. 

HEAT 63 

25.  Radiation   Correction   in  Calorimetry.     26.  The  Beck- 
mann  Thermometer. 
XIII.  Thermometer  Testing. 
XIV.  Temperature  Coefficient  of  Expansion. 

vii 


VI11  CONTENTS. 

PAGE 

XV.  Coefficient  of  Apparent  Expansion  of  a  Liquid. 

XVI.  Coefficient  of  Increase  of  Pressure  of  Air. 

XVII.  Pressure  of  Saturated  Water  Vapor. 

XVIII.  Hygrometry. 

XIX.  Specific  Heat  by  Method  of  Mixture. 

XX.  Ratio  of  Specific  Heats  of  Gases. 

XXI.  Latent  Heat  of  Fusion. 

XXII.  Latent  Heat  of  Vaporization. 

XXIII.  Latent     Heat    of    Vaporization.      Continuous-flow 

method. 

XXIV.  Thermal  Conductivity. 

XXV.  The  Mechanical  Equivalent  of  Heat. 
XXVI.  The  Melting-point  of  an  Alloy. 
XXVII.  Heat  Value  of  a  Solid. 
XXVIII.  Heat  Value  of  a  Gas  or  Liquid. 
XXIX.  Pyrometry. 

SOUND 119 

XXX.  The  Velocity  of  Sound. 
XXXI.  The  Velocity  of  Sound  by  Kundt's  Method. 

LIGHT 124 

27.  Monochromatic  Light.    28.  Rule  of  Signs  for  Spherical 
Mirrors  and  Lenses. 
XXXII.  Photometry. 

XXXIII.  Spectrometer  Measurements. 

XXXIV.  Radius  of  Curvature. 
XXXV.  Focal  Length  of  a  Lens. 

XXXVI.  Lens  Combinations. 

XXXVII.  Magnifying  Power  of  a  Telescope. 

XXXVIII.  Resolving  Power  of  Optical  Instruments. 

XXXIX.  Wave-length  of  Light  by  Diffraction  Grating. 

XL.  Interferometer. 

XLI.  Rotation  of  Plane  of  Polarization. 


ELECTRICITY  AND  MAGNETISM 153 

29.  Resistance  Boxes.  30.  Forms  of  Wheatstone's  Bridge. 
31.  Galvanometers.  32.  Correction  for  Damping  of  a  Bal- 
listic Galvanometer.  33.  Galvanometer  Shunts.  34. 
Standard  Cells.  35.  Device  for  Getting  a  Small  E.  M.  F. 
36.  Double  Commutator.  37.  Relations  between  Electrical 
Units . 


CONTENTS. 


IX 


XLII.  Horizontal    Component    of    the   Earth's    Magnetic 

Field. 

XLIII.   Magnetic  Inclination  or  Dip. 
XLIV.  Measurement      of      Resistance     by     Wheatstone's 

Bridge. 

XLV.  Galvanometer  Resistance  by  Shunt  Method. 
XLVI.  Galvanometer   Resistance  by  Thomson's  Method 
XLVII.  Measurement  of  High  Resistance  (i). 
XLVIII.  Measurement  of  High  Resistance  (2). 
XLIX.   Measurement  of  Low  Resistance  (i). 
L.  Measurement  of  Low  Resistance  (2). 
LI.  Measurement  of  Low  Resistance  (3). 
LI  I.  Comparison    of    Resistances    by  the    Carey-Foster 

Method. 

LIII.   Battery  Resistance  by  Mance's  Method. 
LIV.  Temperature  Coefficient  of  Resistance. 
LV.  Specific  Resistance  of  an  Electrolyte. 
LVI.  Comparison    of    E.    M.    F.'s    by    High    Resistance 

Method. 
LVII.  Comparison    of    E.    M.    F.'s   and    Measurement   of 

Battery  Resistance  by  Condenser  Method. 
LVIII.  Calibration  of  Voltmeter. 
LIX.  Calibration  of  Ammeter. 
LX.  Comparison  of  Capacities  of  Condensers. 
LXI.  Absolute  Determination  of  Capacity. 
LXII.  Coefficients  of  Self-induction  and  of  Mutual  Induc- 
tion. 

LXIII.   Strength  of  a  Magnetic  Field  by  a  Bismuth  Spiral. 
LXIV.  Study  of  a  Ballistic  Galvanometer. 

LXV.  Magnetic  Permeability. 
LXVI.  Magnetic  Hysteresis, 
j  VXTTT      /  (a)  Mechanical  Equivalent  of  Heat. 

\  (b)   Horizontal  Intensity  of  Earth's  Magnetism. 
LXVIII.  Thermoelectric  Currents. 
LXIX.  Elementary    Study  of    Resistance,    Self-induction, 

and  Capacity. 
LXX.  Self-induction,    Mutual    Induction,    and    Capacity, 

Alternating  Currents. 
LXXI.   Dielectric  Constant  of  Liquids. 
LXXII.  Electric  Waves  on  Wires. 


TABLES  

I.  Four-Place   Logarithms. 


235 


CONTENTS. 


II.  Trigmometrical  Functions. 

III.  Reduction  to  Infinitely  Small  Arc. 

IV.  Barometer  Corrections. 

V.   Density  and  Specific  Volume  of  Water. 
VI.   Density  of  Gases. 
VII.   Density,  Specific  Heat,  and  Coefficient  of  Expansion  of 

Metals. 
VIII.   Density,  Specific  Heat,  and  Coefficient  of  Expansion  of 

Miscellaneous    Substances. 
XL   Elastic  Moduli. 

X.   Surface  Tension. 
XI.  Coefficient  of  Viscosity. 
XII.  Specific  Heats  of  Gases. 

XIII.  Vapor  Pressure  of  Water. 

XIV.  Boiling-point  of  Water. 

XV.   Wet  and  Dry  Bulb  Hygrometer. 
XVI.    Vapor  Pressure  of  Mercury. 
XVII.  Melting-points  of  Metals. 
XVIII.  Wave-lengths  of  Light. 
XIX.   Refractive  Indices. 

XX.  Specific  Rotatory  Power. 
XXL  Photometric  Table. 

XXII.   Specific     Resistance     and     Temperature    Coefficient    of 
Metals. 

XXIII.  Specific    Resistance    and    Temperature    Coefficient    of 

Solutions. 

XXIV.  Dielectric  Constants. 

INDEXT 255 


PHYSICAL    MEASUREMENTS. 


INTRODUCTION. 


1.  Purpose  of  Course. 

Intelligent  work  requires  a  clear  perception  of  the  end 
in  view.  It  is  important,  therefore,  to  remember  that  the 
purpose  of  a  course  in  Laboratory  Physics  is  not  only  the 
attainment,  by  personal  experimentation,  of  a  more  definite 
knowledge  of  the  facts  and  principles  of  physics  and  an 
acquaintance  with  the  use  of  measuring  instruments  and 
methods,  but  also  the  acquisition  of  a  scientific  habit  of 
accuracy  and  carefulness  in  observing  and  examining  phe- 
nomena and  drawing  conclusions  therefrom. 

2.  General  Directions. 

Much  time  in  the  laboratory  will  be  wasted  unless  some 
preparation  be  made  before  coming  to  the  laboratory.  The 
purpose  and  general  method  of  the  measurement  to  be  made 
should  be  examined  with  the  aid  of  the  text-book  and  some 
of  the  references  preceding  the  directions.  This  may 
usually  be  done  in  a  few  minutes  at  home,  whereas  it  might 
require  an  hour  or  more  in  a  laboratory  where  a  number 
of  people  are  moving  around. 

The  readings  made  in  the  laboratory  should  always  be 
recorded  in  a  firmly  bound  book  reserved  for  this  purpose 
only,  and  never  on  loose  slips  of  paper  or  in  a  book  that  may 
become  dog-eared  and  untidy.  When,  for  convenience  or 
of  necessity,  two  work  together  at  an  experiment,  each 
should  keep  his  own  notes  of  the  measurements  made,  and, 

i 


2  INTRODUCTION. 

whenever  possible,  each  should  make  a  separate  set  of  read- 
ings for  himself,  and  these  should  be  as  independent  as 
possible. 

No  operation  should  be  performed  or  measurement 
made  unless  the  purpose  and  meaning  of  it  are  understood ; 
otherwise  it  may  be  made  imperfectly  or  some  essential 
part  of  it  may  be  overlooked. 

3.  Reports. 

An  essential  part  of  the  work  is  a  written  report  on  each 
experiment  completed.  This  should  be  handed  in  within  a 
week  after  the  work  is  finished.  In  preparing  the  report 
the  writer  has  to  make  clear  to  himself  the  purpose  and 
bearing  of  each  part  of  the  work  and  examine  critically  the 
value  and  accuracy  of  the  final  result.  This  exercise  is  as 
valuable  as  the  experimental  work  itself.  The  report 
should  be  as  brief  as  possible,  consistently  with  giving  the 
following  information : 

The  purpose  of  the  experiment  (including  the  definition 
of  the  leading  terms,  such  as  coefficient  of  friction,  mechani- 
cal equivalent  of  heat,  etc.) ; 

A  brief  statement  of  the  method  used ; 

A  statement  (tabulated  if  possible)  of  the  observations 
and  readings  made : 

An  outline  of  the  calculation  of  the  final  result  (omit- 
ting the  details  of  the  numerical  work) ; 

A  criticism  of  the  reliability  of  the  result ; 

Brief  answers  to  the  questions  appended  to  the  directions. 

4.  Errors. 

A  perfectly  accurate  experimental  result  is  impossible; 
but  some  estimate  can  usually  be  formed  as  to  the  magni- 
tude of  the  possible  error  and  this  is  frequently  of  the 
greatest  value.  An  experimental  result  of  unknown  reli- 
ability is  often  of  very  little  value.  Hence  an  estimate  of 


ERRORS    OF    OBSERVATION.  3 

the  accuracy  of  a  measurement  is  very  desirable  in  an 
account  of  the  work. 

Inaccuracy  may  arise  from  several  different  causes — 
(i)  errors  of  observation,  due  to  the  inherent  limitations 
of  the  observer's  powers  of  observing  and  judging;  (2) 
instrumental  errors,  arising  from  imperfections  in  the  work 
of  the  instrument  maker  in  constructing  and  subdividing 
the  scale  used  by  the  observer;  (3)  mistakes,  such  as  the 
mistaking  of  an  8  for  a  3  on  a  scale;  (4)  systematic  errors 
due  to  faultiness  in  the  general  method  employed. 

Instrumental  errors  may  be  decreased  by  using  more 
accurate  instruments  or  by  calibrating  the  scales  of  the  instru- 
ments used,  that  is,  ascertaining  and  allowing  for  the  errors 
in  their  graduation.  This  is  frequently  a  difficult  operation 
and  unsuited  for  an  elementary  course.  We  shall,  therefore, 
usually  assume  that  the  accuracy  of  the  instruments  is 
such  that  the  instrumental  errors  are  less  than  the  errors  of 
observation. 

Mistakes  in  reading  can  be  eliminated  by  care  and  repeti- 
tion. Systematic  errors  are  apt  to  arise  when  some  indirect 
method  of  arriving  at  a  result  is  adopted,  a  direct  method 
being  difficult  or  impossible.  For  example,  the  length  of 
a  wave  of  light  cannot  be  measured  directly  and  a  method 
depending  on  diffraction  or  interference  is  usually  employed 
(Exp.  XXXIX).  A  careful  study  of  the  method  used  will 
often  enable  us  to  eliminate  such  errors  by  improving 
the  details  of  the  method,  or,  where  this  cannot  be  done, 
some  estimate  of  the  uneliminated  errors  can  often  be  formed. 

5.  Errors  of  Observation. 

Different  methods  of  estimating  the  magnitude  of  errors  of 
observation  may  be  employed,  the  choice  depending  on  the 
nature  of  the  measurements.  In  many  cases  the  quantity 
can  be  measured  several  times  and  the  mean  taken,  it 
being  probably  more  accurate  than  a  single  observation. 
In  other  cases  circumstances  do  not  permit  repetition  and  a 


4  INTRODUCTION. 

single  observation  must  suffice.  In  either  case  the  observer 
can,  from  the  circumstances  of  the  case,  say  with  a  high  de- 
gree of  probability  that  the  error  cannot  be  greater  than  a 
certain  magnitude.  This  we  shall  call  the  "possible  error" 
of  the  measurement.  It  does  not  strictly  mean  the  greatest 
possible  error,  since  a  greater  error  might  be  theoretically 
possible  but  very  improbable. 

(a)  When  Only  a  Single  Observation  is  Made. — For  example, 
a  liquid,  the  temperature  of  which  is  varying  slowly,  is  kept 
well  stirred  and  the  temperature  is  observed  by  means  of  a 
thermometer  graduated  to  degrees.     The  temperatur    at  a 
certain  time  is  noted  as  being  between  36°  and  37°  and  the 
observer,  estimating  to  o .  i  of  a  division,  records  the  tem- 
perature as  36.3°;  but  he  does  not  trust  his  estimate  closer 
than  o .  i ;  that  is,  he  considers  that  the  real  temperature  may 
be  as  high  as  36  . 4°  or  as  low  as  36 .  2°.     He  therefore  states 
the  temperature  as  36.3°  with  a  possible  error  of  0.1°,  or 
36.3°±o.i°.     The  actual  error  may,  of  course,  be  less  than 
o.  i°;  the  latter  is  only  a  reasonable  estimate  of  the  limit  of 
error  of  observation. 

(b)  When   Several   Different   Observations   of  a   Quantity 
are  Made. — The   mean   of   a  number  of  observations  of  a 
quantity    is  more  trustworthy  than  a  single    reading,  for 
observations  that  are  too  large  are  likely  to  counterbalance 
others    that    are    too    small.     Greater   confidence    can    be 
placed  in  the  mean  when  the  separate  readings  differ  but 
little   from  the  mean  than  when  they  differ  greatly.     The 
average    of   the    differences    between    the    mean    and    the 
separate  readings  is  called  the  mean  deviation.     It  can  be 
shown  (as  indicated  in  §9)  that  when  ten  observations  are 
made,  the  probability  that  the  actual  error  is  greater  than 
the  mean  deviation  is  very  small,  about  i  in  100,  while  if 
15  observations  are  made  it  is  reduced  to  i  in  1000.     Even 
if  only  5  observations  are  made  (which  is  rather  too  small  a 
number)  the  probability  is  only  i  in  15.     Hence,  when  a 
quantity  is  measured  several  times,  the  average  deviation 
may  be  taken  as  a  measure  of  the  possible  error. 


POSSIBLE    ERROR    OF   A    CALCULATED    RESULT.  5 

6.  Possible  Error  of  a  Calculated  Result. 

A  piece  of  laboratory  work  usually  calls  for  the  measure- 
ment of  several  different  quantities  and  the  calculation  of  a 
result  by  some  formula.  Knowing  the  possible  errors  of 
the  separate  quantities  we  can  deduce  the  possible  error 
of  the  result,  but  the  method  will  vary  with  the  nature  of 
the  arithmetical  operations. 

(a)  Possible  Error  of  a  Sum  or  Difference. — The  possible 
error  of  a  sum  or  difference  is  the  sum  of  the  possible  errors 
of  the  separate  quantities,  for  each  possible  error  may  be 
either  positive  or  negative. 

Example. — A  bulb  containing  air  (Exp.  VI)  weighs 
2o.i425g.  ±0.0002  g.  and  after  the  air  has  been  pumped 
out  it  weighs  20.0105  g.  ±o.ooo2g.  Hence  the  weight  of  the 
air  is  o .  1 3  2 o  g.  ±  o .  0004  g.  Since  it  is  sometimes  erroneously 
assumed  that  a  derived  result  must  be  accurate  to  as  high  a 
percentage  as  the  measurements  from  which  it  is  deduced, 
it  should  be  noticed  in  the  above  that,  while  the  separate 
weights  are  found  to  0.001%,  the  weight  of  the  air  is  only 
ascertained  to  0.3%. 

(b)  Possible  Error  of  a  Power. — If  a  measured  quantity 
x  is  in  doubt  by  p  per  cent  (p  being  small),  the  nth  power 
of  x  is  in  doubt  by  up  per  cent.     For 


x\  i±- 

IOO/     I  \  IOO 


squares  and  higher  powers  of  p/ioo  being  neglected. 

Example  of  (a)  and  (b). 

T  =  3.  506  ±  .005  and/  =  2.018  ±  -003.  (Exp.  VII).  What 
is  the  possible  error  of  T2  —  t2?  T2=  12.  2  9  and  since  T  may  be  in 
error  by  1/7  %,  T2  maybe  in  error  by  2/7  %  or  .04.  Hence 
T2  =  12  .  29  ±  .04.  Similarly  £2  =  4.07  ±  .01.  Hence  T2—t2  = 

8.  22  ±  .05. 

(c)  Possible  Error  of  a  Product  or  Quotient. — The  percent- 
age by  which  a  product  or  quotient  is  in  doubt  is  the  sum  of 


6  INTRODUCTION. 

the  percentages   by  which   the   separate   quantities   are  in 
doubt.     For  if  the  quantities  be 

/         P  \  /         <1  \ 

x(  i  ± )  and  y{  i  ± I 

V       ioo/  \       ioo/ 

their  product  is 

x(I±JL\J1±A\^xyL±P^} 

\  IOO/          \  IOO/  \  IOO/ 

and  their  quotient  is 


p/ioo  and  q/ioo  being  assumed  small.  It  is  evident  that 
a  similar  statement  applies  to  any  number  of  products  and 
quotients. 

Example  of  (b)  and  (c) . 

The  diameter  of  a  sphere  (Exp.  Ill)  is  measured  by 
a  vernier  caliper  and  found  to  be  i .  586  cm.,  but  the  vernier 
only  reads  to  1/50  mm.;  so  the  possible  error  is  .002  cm.  or 
1/8  of  i%.  The  sphere  is  weighed  in  a  balance  such  that  i 
mg.  added  to  one  pan  does  not  cause  an  observable 
change  of  the  pointer,  while  2  mg.  does,  and  the  weight  is, 
therefore,  16.344  g.  with  a  possible  error  of  .002  g.  or 
1/80  %.  The  calculated  value  of  the  density  is  7  . 827 ;  but  the 
volume  may  be  in  error  by  3/8%  and  the  mass  by  1/80%. 
Hence  the  density  may  be  in  error  by  3  /  8  + 1  /  80  % ,  or  practi- 
cally 3/8%.  Hence  the  proper  statement  of  the  density 
is  7  . 83  with  a  possible  error  of  .03  or  7  . 83  ±  .  03. 

7.  General  Method  for  the  Possible  Error  of  a  Result. 

The  above  rules  for  sums,  differences,  powers,  products, 
and  quotients  will  usually  suffice  for  finding  the  possible 
error  of  a  result  calculated  from  the  measurements  of 
several  quantities.  But  when  several  of  these  operations 
are  combined,  or  when  the  formula  for  calculation  contains 


GENERAL    METHOD    FOR    THE    POSSIBLE    ERROR.  7 

one  of  the  quantities  more  than  once,  the  effects  of  the 
several  errors  may  be  difficult  to  trace  by  these  means.  The 
following  general  method  is  always  applicable.  It  may  be 
carried  out  by  simple  arithmetic,  but  is  simplified  by  an 
elementary  use  of  the  calculus. 

To  find  to  what  extent  the  possible  error  in  one  of  the 
quantities  affects  the  result,  we  may  calculate  the  result 
assuming  all  the  quantities  to  be  quite  accurate  and  then 
repeat  the  calculation  after  changing  one  of  the  quantities 
by  its  possible  error.  The  difference  in  the  result  will  be  the 
effect  sought.  If  we  do  the  same  for  each  of  the  other 
quantities,  the  final  possible  error  of  the  result  will  be  the 
sum  (without  regard  to  sign)  of  the  parts  due  to  the  separate 
quantities. 

This,  however,  is  equivalent  to  differentiating  the  whole 
expression,  first  with  regard  to  one  quantity,  then  with 
regard  to  a  second  and  so  on  and  finally  adding  the  partial 
differentials.  It  will  be  seen  from  the  following  examples 
that  the  process  is  much  simplified  by  taking  the  logarithm 
of  the  whole  formula  before  differentiating. 

(i)  Time  of  Vibration  of  a  Pendulum  (Exp.  VII).  —  If  in 
time  T  a  pendulum  makes  n  fewer  vibrations  than  the 
pendulum  of  a  clock  that  beats  seconds  and  if  t  is  the  time 
of  a  single  vibration, 

T 

t  =  T-n 
Taking  logarithms, 

log  t  =  log  T-log(T-n) 
Hence  by  differentiating, 

&_sr     sr 

t      T     T-n 


This  means  that  if   T  be  changed  by  a  small  quantity,  ST 
the  consequent  change,  Bt,  in  t  is  given  by  the  formula.     If 


8  INTRODUCTION. 

the  possible  error  of  T  be  2  seconds,  by  putting  8T  =  ±  2 
the  value  of  $t  will  be  the  possible  error  of  t.  If  T  be  862 
seconds  and  n  be  17, 

&  17X2 


This  indicates  one  of  the  advantages  of  taking  logarithms. 
It  gives  us  at  once  the  ratio  of  $t  to  t,  or  (multiplied  by  100) 
the  percentage  by  which  t  is  in  doubt. 

(2)  Specific  Heat  by  the  Method  of  Mixture  (Exp.  XIX).— 
Let  r  =  95°  be  the  initial  temperature  of  the  specimen, 
20  =  25°  that  of  the  water,  and  let  t  =  45°  be  the  final  tempera- 
ture of  the  mixture,  and  let  the  possible  error  of  each  ther- 
mometer reading  be  o.  2°.  The  formula  for  calculation  is 


^ 


M(T-t) 


We  shall  consider  how  far  the  possible  errors  in  the  ther- 
mometer readings  affect  x,  leaving  the  consideration  of  the 
other  terms  (the  errors  of  which  are  likely  to  be  much  smaller) 
to  the  reader. 

log  x=log(m+mls)  +log(t—t0)  —logM  —  log(T-t) 

Proceeding  as  in  (i)  above,  we  find  the  effects  of  the  possible 
errors  of  T,  t0,  and  /  respectively  as  follows : 

$x  SjT  o.  2 

=  0.4% 


T  —  t  50 

&  0.2 


/-/„  20 


=  1  .O 

70 


Total  =  2. 8% 

This  example  will  show  a  second  advantage  in  the  method  of 
taking  logarithms.  It  separates  the  various  terms  and  so 
simplifies  the  process. 


OF   A    MEAN.  9 

8.  Some  General  Notes  on  Errors. 

The  statement  of  a  possible  error  should  contain  only  one 
significant  figure.  (A  zero  that  serves  only  to  fix  the  decimal 
point,  such  as  the  zeros  in  0.0026,  is  not  a  significant 
figure).  Thus  in  the  last  example  in  §6,  3/8%  of  7.83  is 
0.0293,  which  shows  that  the  second  decimal  place  in  7.83 
is  in  doubt  by  3.  Hence  it  would  be  superfluous  to  add 
figures  to  show  that  the  third  and  fourth  decimal  places  are 
also  in  doubt. 

Measurements  sometimes  seem  so  accurate  that  one  is 
tempted  so  say  that  "the  possible  error  is  practically  zero 
and  need  not  be  considered."  This  is  never  literally  true. 
One  factor  may  be  so  accurately  determined,  compared  with 
other  factors,  that  the  effect  of  its  possible  error  on  the  result 
might  seem  to  be  negligible ;  but  only  a  calculation  can  show 
this  and  the  calculation  will  frequently  show  the  opposite. 
An  illustration  occurs  in  the  first  example  of  §6,  considered 
in  connection  with  the  other  measurements  required  to 
determine  the  density  of  air  in  Exp.  III. 

The  consideration  of  possible  errors  is  of  great  importance 
in  deciding  what  care  need  be  expended  in  determining  the 
various  factors  in  a  complex  measurement  and  what  are  the 
best  conditions  for  obtaining  an  accurate  result.  This 
applies  more  especially  to  advanced  and  difficult  measure- 
ments, but  illustrations  will  occur  in  this  book.  (Exp.  XIX.) 
But,  as  one  of  the  purposes  of  this  course  is  to  teach  the  most 
exact  use  of  the  measuring  instruments,  measurements 
should  usually  be  made  as  accurately  as  the  instruments 
will  permit. 

9.  "Probable  Error"  of  a  Mean. 

There  is  another  method  of  indicating  the  reliability  of 
measurements  which  possesses  some  advantages  over  the 
one  that  we  have  explained,  though  it  is  not  so  generally 
applicable.  When  a  large  number  of  observations  of  a 
quantity  have  been  made,  we  can,  by  means  of  formulas 


10  INTRODUCTION. 

deduced  from  the  mathematical  Theory  of  Probability,  cal- 
culate the  probability  that  the  mean  is  not  in  error  by  more 
than  a  given  amount.  When  a  coin  is  tossed  up  it  is  an  even 
chance  whether  it  will  come  down  a  head  or  a  tail;  the 
chance  or  probability  of  its  being  a  head  is,  therefore,  i  in  2 
or  1/2.  Now  the  "probable  error"  of  the  mean  of  a  num- 
ber of  readings  is  defined  as  a  magnitude  such  that  it  is  an 
even  chance  whether  the  error  is  greater  or  whether  it  is 
less  than  this  magnitude.  In  other  words,  the  probabil- 
ity of  the  error  exceeding  the  "probable  error"  is  1/2. 
One  formula  for  calculating  the  "probable  error"  is  the 
following : 

average  deviation 
#=0.84—^ 

A/number  of  observations 

This  method  is  useful  as  a  method  of  indicating  the  reli- 
ability of  measurements  when  each  of  all  the  quantities  that 
occur  in  the  experiment  can  be  measured  several  times. 
For  when  each  mean  and  its  "probable  error"  has  been 
found  we  can  calculate  the  "probable  error"  of  the  final 
result.  For  further  details  we  shall  refer  the  reader  to  other 
works  (e.  g.,  Merriman's  "Least  Squares").  We  shall  not 
have  frequent  occasion  to  refer  to  "  probable  errors,"  since  in 
most  cases  some  of  the  quantities  that  have  to  be  determined 
cannot  be  measured  more  than  once. 

In  justification  of  the  use  of  the  mean  deviation  as  a 
measure  of  the  possible  error,  we  may  note  that  by  the  above 
formula  for  e,  when  10  observations  have  been  made,  the 
mean  deviation  equals  3  . 6  <?.  Now  a  reference  to  tables 
of  the  probability  of  errors  (e.  g.,  Smithsonian  Tables)  shows 
that  the  probability  of  an  error  greater  than  3.8  e  is  about  i 
in  100. 

10.  Limits  to  Calculations. 

By  the  above  methods  the  possible  error  in  any  calcula- 
tion from  experimental  quantities  may  be  deduced.  The 
magnitude  of  the  possible  error  in  any  calculation  indicates 


PLOTTING    OF    CURVES.  II 

how  far  it  is  useful  and  desirable  to  carry  the  calculation. 
A  calculation  should  be  carried  as  far  as,  but  not  farther  than, 
the  first  doubtful  figure.  This  rule  must  be  applied  not  only 
to  the  calculation  of  the  final  result,  but  also  to  each  inter- 
mediate step.  When  a  calculation  is  carried  too  far,  useless 
and  very  unscientific  labor  is  expended,  and  when  it  is  not 
carried  far  enough  very  absurd  results  are  often  obtained. 

In  addition  and  subtraction  a  place  of  decimals  that  is 
doubtful  in  any  one  of  the  quantities  is  doubtful  in  the 
result. 

In  multiplication  and  division  (performed  in  the  ordinary 
way)  decimal  places  that  have  not  been  determined  are 
usually  filled  up  by  zeros.  'Any  figure  in  the  result  that 
would  be  altered  by  changing  one  of  these  zeros  to  5  is 
doubtful. 

ii.  Notation  of  Very  Large  and  Very  Small  Numbers. 

Partly  to  save  space  and  partly  to  indicate  at  once  the 
magnitude  of  very  large' or  very  small  numbers,  the  following 
notation  is  used.  The  digits  are  written  down  and  a  decimal 
point  placed  after  the  first  and  its  position  in  the  scale 
indicated  by  multiplying  by  some  power  of  10.  Thus 
42140000  is  written  4.2i4Xio7  and  .00000588  is  written 
5.88X10-®.  This  also  enables  us  to  abbreviate  the  multi- 
plication and  division  of  such  numbers.  Thus  42 140000  X 
.00000588  is  the  same  as  4.  214 X  5-88X  10  and  42140000-7- 
.00000588  is  the  same  as  (4.  214-=-  5. 88)  X  io13. 

12.  Plotting  of  Curves. 

It  is  assumed  that  the  general  method  of  the  represen- 
tation of  the  connection  between  two  related  quantities  by 
means  of  a  curve  is  familiar  to  the  reader  from  his  work  in 
Graphical  Algebra  or  elsewhere.  Attention  may  be  called 
to  the  following  points: 

i.  Mark  experimental  points  clearly  by  crosses  or  circles 
surrounding  the  points. 


12  INTRODUCTION. 

2.  The  curve  should  be  drawn  so  as  to  strike  an  average 
path  among  the  points;  it  does  not  have  to  pass  through  even 
one  point. 

3.  Abscissas  may  be  drawn  to  any  scale  and  ordinates  to 
any  scale.     Record  the  main  division  of  the  scales  along 
each  axis.      (Do  not  record  the  individual  observations.) 

4.  For  convenience,  all  the  abscissas  may  be  diminished 
by  the  same  amount  before  plotting,  and  the  same  is  true  of 
ordinates. 

5.  Ordinates  and  abscissas  should  be  drawn  to  such  scales 
that  the  curve  occupies  a  large  part  of  the  paper. 

6.  Curves  should  be  drawn  carefully  and  neatly  by  means 
of  curve  forms. 


MECHANICS. 

13.  The  Use  of  a  Vernier. 

The  vernier  is  a  contrivance  for  reading  to  fractions  of  the 
units  in  which  scales  are  graduated.  It  is  a  second  scale 
parallel  to  the  main  scale  of  the  instrument  and  so  divided 
that  n  of  its  units  equal  n  —  i  or  n  -f  i  of  the  units  of  the  scale. 
If  5  is  the  length  of  a  scale  unit  and  v  that  of  a  vernier  unit, 
in  the  first  case 

nv=(n—  i)s  or  s  —  v  —  s-^-n; 
in  the  second  case 

nv=(n  +  i)s  or  v  —  s  =  s  +  n. 

Hence  the  unit  of  the  vernier  is  less  or  greater  than  that  of 
the  scale  by  one-wth  of  a  scale  unit. 

If  we  did  not  have  a  vernier  there  would  be  something  in 
the  nature  of  an  index  to  indicate  what  division  of  the  scale 
should  be  read  in  making  a  certain  measurement  and 
fractions  would  be  estimated  by  eye.  The  zero  of  the  vernier 
is  taken  as  such  an  index,  the  whole  number  of  scale  divisions 
being  the  number  just  below 
the  zero  of  the  vernier,  while 
the  fraction  of  a  scale  division 


is  determined  with  the  vernier.  A|J  I  I  IJ  I  I  I  I  ,|0 

If    the    wth     division     of     the  FlG    T 

vernier  coincides  with  the  scale 

division,  the  zero  of  the  vernier  must  be  m  wths  of  a  scale 

unit  from  the  scale  division  just  below  it.     Thus  the  use  of 

a  vernier  divided  into  n  parts  is  equivalent  to  subdividing 

the  scale  unit  into  n  parts. 

If   no    vernier    division    exactly    coincides    with    a    scale 
division  there  will  be  two  vernier  divisions  nearly  coincident 


MECHANICS. 


0 


FIG.  2. 


with  scale  divisions,  and  one  can  often  estimate  fractions 
of  the  fraction  given  by  the  vernier.  In  figure  i  the 
whole  number  of  scale  divisions  is  5.2,  and  evidently  the 
third  and  fourth  vernier  divisions  most  nearly  coincide  with 
scale  divisions.  Since  the  third  division  is 
somewhat  nearer  a  coincidence,  we  may  call 
the  fraction  3 . 3  tenths,  or,  the  full  reading 
will  be  5  .  233. 

14.  Vernier  Caliper. 

The  vernier  caliper  consists  of  a  straight 
graduated  bar,  and  two  jaws  at  right  angles 
to  it,  one  of  which  is  fixed  while  the  other 
is  movable.  The  position  of  the  movable 
jaw  can  be  accurately  determined  by  means 
of  the  scale  and  a  vernier  which  should  read 
zero  when  the  jaws  are  in  contact.  (If  this  be 
not  the  case,  allowance  must  be  made  for  the  zero  reading) . 

15.  Micrometer  Caliper. 

The  micrometer  caliper  is  a  U-shaped  piece  of  metal  in 
one  arm  of  which  is  a  steel  plug  with  a  carefully  planed  face 
and  through  the  other  arm  of  which  passes  a  screw  with  a 
plane  end  parallel  and  opposite  to  that  of  the 
screw.  A  linear  scale  on  the  frame  reads  zero 
approximately  when  the  plug  and  screw  are  in 
contact,  and  its  reading  in  any  other  position 
indicates  the  whole  number  of  turns  of  the 
screw  and  consequently  the  number  of  mm. 
(or  1/2  mm.  or  1/40  in.  as  the  pitch  of  the 
screw  may  be)  between  the  screw  and  the  plug. 
Fractions  of  a  turn  are  read  on  the  divided 
head.  As  contact  approaches,  the  screw  should  be  turned 
with  a  very  light  touch  and  the  same  force  used  for 
different  contacts.  Some  micrometers  are  provided  with  a 
ratchet  head  which  permits  only  a  definite,  moderate 
pressure. 


MICROMETER    MICROSCOPE.  15 

16.  Micrometer  Microscope. 

The  micrometer  microscope  is  a  microscope  with  cross- 
hairs at  the  focus.  In  one  type  of  instrument  these  cross- 
hairs are  movable  by  a  micrometer  screw.  In  the  other 
and  more  common  type  the  whole  microscope  is  moved  by 
a  micrometer  screw  (see  Fig.  4).  The  most  elaborate 
instruments  have  both  movements.  The  rotations  of  the 
screw  are  read  on  a  fixed  linear  scale  while  the  fraction  of  a 
rotation  is  read  by  a  circular  scale  attached  to  the  screw, 
and  thus  the  amount  of  movement  is  ascertained  if  the 
pitch  of  the  screw  is  known.  The  pitch  is  best  determined 
by  reading  a  length  on  a  reliable  scale  placed  in  the  field  of 
view. 

17.  Comparator. 

The  comparator  consists  essentially  of  a  pair  of  micro- 
scopes movable  along  a  horizontal  bar  to  which  they  are  at 
right  angles.  The  length  to  be  measured  is  placed  under  the 


FIG.  4. 

microscopes.  The  eye-piece  of  each  microscope  is  first  focused 
clearly  on  the  cross-hairs  and  the  whole  microscope  focused 
without  parallax  on  the  point  to  be  observed,  so  that  the 
image  of  the  point  coincides  with  the  intersection  of  the 
cross-hairs.  The  object  is  then  removed  and  a  good  scale 
put  in  its  place,  and  a  reading  of  the  scale  gives  the  required 
length,  this  reading  being  facilitated  by  the  use  of  microm- 
eter screws. 


1 6  MECHANICS. 

18.  Spherometer. 

The  Spherometer  is  an  instrument  with  four  legs,  three 
of  which  form  the  vertices  of  an  equilateral  triangle,  while 
the  fourth  is  at  the  center  of  the  triangle.  The  fourth  leg 
can  be  screwed  up  and  down  and  the  distance  of  its  extrem- 
ity from  the  plane  of  the  extremities  of  the  other  three  legs 
can  be  accurately  measured  by  means  of  a  linear  scale 
attached  to  the  fixed  legs  and  a  circular  scale  attached  to 
the  movable  leg.  The  linear  scale  gives  the  number  of 
complete  turns  of  the  screw  and  the  circular  scale  the  frac- 
tion of  a  turn.  These  scales  are  read  when  the  screw  makes 
contact  with  an  object  placed  beneath. 

The  position  of  contact  may  be  determined  by  noticing 
that  the  screw  turns  very  easily  for  a  fraction  of  a  turn  just 
after  contact  begins.  This  is  due  to  reduced  friction  in  the 
bearing,  owing  to  the  weight  of  the  screw  falling  on  the 
body  in  contact  and  to  the  back-lash  of  the  screw,  the 
frame  not  yet  being  raised.  The  screw  should 
be  lowered  until  it  thus  begins  to  turn  very 
easily;  it  should  then  be  turned  back  again 
until  it  just  again  begins  to  turn  hard  and  a 
reading  then  made.  A  less  sensitive  method 
is  to  turn  the  screw  down  until  the  instrument 
is  felt  to  rock  or  wobble  and  make  a  reading; 
then  raise  the  screw  until  rocking  just  ceases, 
make  another  reading  and  take  the  mean  of 
FIG.  5.  "the  two  as  the  contact  reading.  In  some 
spherometers  the  end  of  the  screw  on  making 
contact  raises  two  levers  arranged  to  greatly  magnify  the 
motion.  The  screw  is  lowered  until  the  top  lever  comes  to 
some  definite  position,  for  instance,  with  the  end  opposite 
to  a  stud  in  the  frame  of  the  instrument. 

The  zero  reading  is  the  reading  of  the  scales  when  the  end 
of  the  screw  is  in  the  plane  of  the  ends  of  the  legs,  and  is 
so  called  because  the  instrument  is  most  frequently  used  to 
determine  distances  above  or  below  this  plane.  It  may  be 


THE    DIVIDING    ENGINE.  17 

obtained  by  placing  the  instrument  on  a  very  plane  plate  of 
glass.  Several  zero  readings  should  be  made  before  and 
after  making  readings  with  the  object  in  position  under  the 
screw,  for  the  zero  reading  is  likely  to  change  from  slight 
disturbances  of  the  adjustment  of  the  instrument  and  ther- 
mal expansion  due  to  the  heat  of  the  hand.  If  the  mean  of 
the  zero  readings  made  after  reading  on  an  object  should  be 
decidedly  different  from  the  mean  of  those  made  before, 
readings  should  be  again  made  on  the  object  and  the  first 
zero  readings  discarded.  After  each  reading,  the  screw 
should  be  turned  up  through  at  least  a  quarter  revolution, 
that  the  readings  may  be  entirely  independent. 

The  unit  of  the  linear  scale  may  be  obtained  by  com- 
parison with  a  standard  steel  scale.  If  the  plane  of  the 
circular  scale  is  not  exactly  perpendicular  to  the  axis,  the 
linear  scale  may  not  give  the  correct  number  of  turns.  It 
is  therefore  best,  as  a  check  upon  the  linear  scale  readings, 
to  count  the  number  of  rotations. 

19.  The  Dividing  Engine. 

The  dividing  engine  received  its  name  from  its  being 
originally  made  to  subdivide  scales,  diffraction-gratings, 
etc.  An  equally  frequent  use  of  the  instrument  is  for  the 
accurate  measurement  and  calibration  of  scales,  gratings, 
etc. 

It  consists  essentially  of  (i)  a  very  carefully  made  hori- 
zontal screw,  the  ends  of  which  are  so  supported  that  the 
screw  is  free  to  rotate,  but  not  to  advance  or  recede;  (2)  a 
nut  movable  on  the  screw  and  bearing  against  (3)  a  plat- 
form movable  along  a  track  which  is  parallel  to  the  screw; 
(4)  a  micrometer  microscope  which  in  some  instruments  is 
held  in  a  support  movable  along  a  rail  on  the  same  bed 
plate  as  the  track,  but  in  other  instruments  is  carried  on  the 
platform;  (5)  dividing  gear  for  making  scales,  etc.;  (6)  a 
divided  circular  scale  attached  to  the  screw  with  a  vernier 
attached  to  the  bed  plate. 


1 8  MECHANICS. 

If,  by  means  of  the  circular  scale  and  vernier,  the  rota- 
tion of  the  screw  can  be  read  to  i  in  a  very  large  number, 
say  in  i  in  1,000,  then,  since  the  nut  moves  a  distance  equal  to 
the  "pitch"  of  the  screw  (measured  parallel  to  the  axis) 
when  the  screw  is  given  one  complete  rotation,  it  follows 
that  the  movement  of  the  nut  and  platform  can  be 
read  to  a  correspondingly  small  fraction  of  the  pitch  of 
the  screw. 

The  object  whose  length  is  to  be  measured  is  placed  on 
the  movable  platform  (in  the  case  of  instruments  of  the  first 
type  mentioned  under  (4)  above) .  The  microscope  is  focused 
on  one  end  of  the  length  to  be  measured,  so  that  the  inter- 
section of  the  cross-hair  coincides  with  that  end.  By  turn- 
ing the  screw  until  the  movement  of  the  platform  brings  the 
other  end  of  the  length  to  be  measured  into  coincidence 
with  the  intersection  of  the  cross-hairs  and  observing  the 
number  of  turns  and  parts  of  a  turn,  the  length  of  the  object 
in  terms  of  the  pitch  of  the  screw  as  unit  is  obtained.  The 
true  pitch  of  the  screw  must  itself  be  obtained  by  com- 
paring it  by  the  same  method  with  some  accurately  known 
length,  such  as  a  length  on  a  standard  meter. 

The  adjustment  of  the  microscope  consists  of  two  steps: 
(i)  the  eye-piece  must  be  focused  on  the  cross-hairs  (but 
the  eye-piece  must  not  be  taken  out  lest  the  cross-hairs  be 
injured) ;  (2)  the  whole  microscope  must  be  moved  toward 
or  away  from  the  object  until  it  is  seen  without  parallax,  i.  e., 
until  the  relative  position  of  the  cross-hairs  and  the  image 
of  the  object  is  not  changed  by  shifting  the  eye  sidewise. 
The  length  to  be  measured  must  then  be  placed  parallel 
to  the  screw.  This  is  attained  when,  by  rotation  of  the  screw, 
the  image  of  each  end  can  be  brought  to  coincidence  with 
the  intersection  of  the  cross-hairs.  One  of  the  most  fre- 
quent sources  of  error,  in  using  a  measuring  instrument 
on  the  screw  principle,  is  back-lash  or  lost  motion.  To 
avoid  this  the  screw  should  always  be  turned  in  the  same 
direction  during  a  measurement.  In  many  dividing 
engines  back-lash  is  impossible  because  the  motion  of  the 


CATHETOMETER.  19 

screw  cannot  be  reversed,  the  platform  can  only  be  moved 
in  the  reverse  direction  by  unclasping  the  nut  (which,  for 
this  purpose,  is  a  split  nut  held  together  by  a  clasp  and 
spring) . 

Usually  the  handle  for  turning  the  screw  is  not  attached 
to  the  screw  or  circular  scale,  but  to  a  separate  disk  rotating 
co-axially  with  the  screw.  The  motion  of  the  handle  in 
one  direction  is  communicated  to  the  screw  by  a  ratchet; 
when  the  handle  is  reversed  the  ratchet  slips  freely.  As  an 
aid  to  counting  the  number  of  turns  of  the  screw  in  measuring 
a  considerable  length,  two  detents  are  sometimes  geared  to 
the  screw  in  such  a  way  that  only  a  definite  number  of  turns 
can  be  given  to  the  screw  at  a  time,  after  which  the  handle 
must  be  turned  back  for  beginning  a  new  number  of  turns. 

A  more  complete  description  of  the  dividing  engine  will 
be  found  in  Stewart  and  Gee,  I,  §16. 

20.  Cathetometer. 

The  cathetometer  is  a  vertical  pillar  supported  on  a  tripod 
and  leveling  screws,  and  capable  of  rotation  about  its  axis; 
the  pillar  is  graduated  and  a  horizontal  telescope  with  cross- 
hairs is  borne  by  a  carriage  that  travels  on  the  pillar  and 
can  be  clamped  at  any  desired  position.  A  slow-motion 
screw  serves  for  accurate  adjustment  of  the  position  of  the 
telescope. 

Adjustments.*  (i)  The  intersection  of  the  cross-hairs, 
X,  must  be  in  the  optical  axis  of  the  telescope.  To  secure 
this,  focus  X  on  some  mark,  rotate  the  telescope  about  its 
own  axis  and  see  whether  X  remains  on  the  mark.  If  not, 
the  adjusting  screws  of  the  cross-hairs  must  be  changed 
until  this  is  attained. 

(2)  The  level  must  be  properly  adjusted.  Level  the 
telescope  until  the  bubble  comes  to  the  center  of  the  scale. 
Turn  the  level  end  for  end.  If  the  bubble  does  not  come  to 

*  Adjustments  (i)  and  (2)  are  not  usually  required  and  should  not  be  made 
without  the  advice  of  the  instructor. 


20  MECHANICS. 

the  same  position,  the  level  must  be  adjusted  until  it  will 
stand  this  test. 

(3)  The  scale   must   be   vertical.     If    there    are    separate 
levels  for  the  shaft,  this  is  readily  attained.      If  there  is  but 
one  level  for  telescope  and  shaft,  this  and  the  next  adjust- 
ment must  be  made  simultaneously. 

(4)  The    telescope    must    be    perpendicular    to    the  scale. 
The  top  of  the  scale,   T,  may  be  regarded  as  having  two 
degrees  of  freedom — first,  parallel  to  the  line  of  two  level- 
ing screws  of  the  base,  A  and  B;  second,  in  a  line  through 
the  third  leveling  screw,   C,   perpendicular  to  AB.     If  A 
and  B  be  screwed  equal  amounts  in  opposite  directions,  T 
will  move  parallel  to  A  B.     If  C  only  be  turned,    T  will 
move  perpendicular  to  AB. 

First  make  the  telescope  horizontal  and  parallel  to  AB. 
Turn  the  shaft  through  180°.  It  is  easily  seen  that  if  the 
telescope  makes  an  angle  a  with  the  normal  to  the  scale, 
turning  the  scale  through  180°  will  cause  the  telescope  to 
make  an  angle  2a  with  its  former  direction.  Hence,  with 
the  leveling  screw  of  the  telescope,  correct  half  the  error 
in  the  level,  and,  by  turning  A  and  B  equally  in  opposite 
directions,  correct  the  remainder.  Turn  the  telescope  to 
the  first  position  and  repeat  the  above  adjustments,  then  to 
the  second  and  continue  as  often  as  is  necessary.  Then  turn 
the  telescope  normal  to  AB  and  adjust  by  C.  When  the 
adjustment  is  complete,  turning  the  shaft  through  any 
angle  will  not  alter  the  position  of  the  bubble. 

Unless  the  cathetometer  is  on  a  perfectly  immovable  sup- 
port, perfect  adjustment  is  not  possible  and  too  much  time 
should  not  be  spent  in  adjusting,  provided  the  telescope  is 
accurately  level  at  each  reading. 

The  eye-piece  of  the  telescope  is  focused  (but  not  removed) 
until  the  cross-hairs  seem  perfectly  distinct  and  the  focus 
of  the  objective  changed  until  the  object  is  seen '  very 
distinctly  and  without  parallax,  i.  e.,  with  no  relative 
motion  with  respect  to  the  cross-hairs  when  the  eye  is 
moved  about. 


BAROMETER.  2 1 

21.  Barometer. 

Text-book  of  Physics  (Duff},  pp.  157-158;  Watson's  Physics,  pp. 
150-154;  Ames'  General  Physics,  pp.  176-177;  Crew's  Physics, 
pp. 165-166. 

If  the  barometer  is  of  Fortin's  cistern  form,  the  cistern  is 
raised  or  lowered  by  means  of  the  screw  at  the  bottom 
until  the  mercury  just  meets  an  ivory  stud  near  the  side  of 
the  cistern.  A  collar  to  which  is  attached  a  vernier  is  so 
placed  that  the  top  of  the  meniscus  of  the  mercury  column 
is  tangent  to  the  plane  of  the  two  lower  edges.  The  height 
of  the  barometer  should  be  reduced  to  zero  by  the  formula 

h0  =  h(i  —  .  00016  2t) 

where  h  is  the  observed  height,  h0  the  height  at  o°  and  /the 
temperature  Centigrade.  For  the  expansion  of  the  mercury 
will  increase  the  height  in  the  ratio  (i .  +  .000181*)  and  the 
expansion  of  the  brass  scale  will  reduce  the  apparent  height 
in  the  ratio  (i  -  .000019*)  (Table  VII). 

The  siphon  barometer  has  two  scales,  graduated  on  the 
glass  tube,  in  opposite  directions,  from  a  common  zero. 
The  length  of  the  mercury  column  is  obviously  the  sum  of 
the  readings  of  the  mercury  levels  in  the  two  tubes.  Since 
the  coefficient  of  linear  expansion  of  glass  is  only  about 
0.000008,  the  correction  formula  becomes 
h0  =  h(i-  .000173*). 

Since  the  mercury  may  adhere  to  the  glass  to  some  extent, 
barometer  tubes  should  be  tapped  gently  before  reading. 

22.  The  Balance. 

Kohlrausch,    §§7-11;    Watson's   Practical  Physics,  §§25,  26. 
Weighing  by  a  Sensitive  Balance. — On  first  using  a  sensitive 
balance   note   the   position,    purpose,    and  structure  of  the 
following  parts: — 

The  beam,  The  knife-edges  and  planes, 

The  pointer,  The  arrestment, 

The  pillar,  The  rider-arms, 

The  pan-supports. 


22  MECHANICS. 

By  the  sensibility  of  a  balance  is  meant  the  amount  of  de- 
flection of  the  beam  produced  by  a  given  small  weight. 
Consider  how  the  sensibility  depends  on  (i)  the  length  of 
the  beam,  (2)  the  weight  of  the  beam  and  pans,  (3)  the 
distance  of  the  point  of  suspension  of  the  beam  from  its 
center  of  gravity.  Will  the  sensibility  of  a  certain  balance 
be  different  with  different  loads  on  the  pans  and  why? 
How  can  the  sensibility  be  varied  with  a  given  load?  (See 
references.) 

Precautions  in  Use  of  Balance. 

1.  Note  the  maximum  load  that  may  be  placed  on  the 
balance  and  take  care  not  to  exceed  it. 

2.  Always  lift   the  beam  from  the  knife  e<Jges  before  in 
any  way  altering  the  load  on  the  pans. 

3.  Do  not  stop  the  swinging  of  the  balance  with  a  jerk. 
It  is  best  to  stop  it  when  the  pointer  is  vertical. 

4.  To  set  the  beam  in  vibration,  do  not  touch  it  with  the 
hand,  but  taise  and  lower  the  arrestment. 

5.  Place  the  large  weights  in  the  center  of  the  pan. 

6.  Make  final  weighings  with  the  case  closed. 

7.  Replace  all  weights  in  their  proper  place  in  the  box 
when  they  are  not  actually  in  use.     Do  not  use  weights 
from  different  boxes. 

8.  Do  not  place  anything  in  contact  with  a  pan  that  is 
liable  to  injure  it. 

9.  Avoid,  if  possible,  weighing  a  hot  body. 

10.  Never  handle  the  weights  with  the  fingers,  as  this 
may  change  some  of  the  weights  appreciably.  Always  use 
the  pincers. 

Notice  the  dimensions  of  the  weights  in  the  box,  e.  g., 
50  g.,  20  g.,  10  g.,  10  g.,  5  g.,  2  g.,  i  g.,  i  g.,  etc.  Instead  of 
weights  of  0.005  £•>  °-  °°2  g->  o.ooi  g.,  o.ooi  g.,  it  is  customary 
to  use  a  rider  of  o .  01  g.,  which  can  be  placed  on  the  beam  at 
various  distances  from  the  center.  The  beam  is  for  this  pur- 
pose graduated  into  10  divisions,  which  may  be  still  further 
subdivided.  Thus  the  o.oio  g.  rider  placed  at  the  division 
4  of  the  beam  is  equivalent  to  o .  004  g.  placed  on  the  pan. 


THE    BALANCE.  23 

The  zero-point  of  the  balance  is  the  position  on  the  scale 
behind  the  pointer  at  which,  the  pans  being  empty,  the 
pointer  would  ultimately  come  to  rest;  it  must  not  be  con- 
fused with  the  zero  of  the  scale.  As  much  time  would  be 
wasted  in  always  waiting  for  the  pointer  to  come  to  rest, 
the  zero  of  the  balance  is  best  obtained  from  the  swings  of 
the  pointer.  For  this  purpose,  readings  of  the  successive 
"turning-points"  are  made  as  follows — three  successive 
turning-points  on  the  right  and  the  two  intermediate  ones 
on  the  left,  or  vice  versa;  e.  g., 

Turning  points. 
L.  R. 

-1-3  +  2-1 

—  I.I  -f  2.  O 

—  I  .O 

Mean,  —1.13  +2.05 

-1.13 

Zero-point  =  +0.92^-2  = +0.46. 

By  taking  an  odd  number  of  successive  turning-points 
on  one  side  and  the  intermediate  even  number  on  the  other 
side  and  then  averaging  each  set,  we  eliminate  the  effect  of 
the  gf  adual  decrease  of  amplitude  of  the  swing. 

The  resting-point  of  the  balance  with  any  loads  on  the 
pans  is  the  point  at  which  the  pointer  would  ultimately 
come  to  rest,  and  is  found  in  the  same  way  as  the  zero-point. 
If  the  resting-point  should  happen  to  be  the  same  as  the 
zero-point,  the  weight  of  the  body  on 
one  pan  is  immediately  found  by  the  ~ 
weights  on  the  other  pan  and  the  posi- 
tion of  the  rider.  Usually,  however,  this 

will  not  be  so.  With  the  rider  at  a  suitable  division,  find  the 
resting-point  on  one  side  of  the  zero-point,  and  then,  after 
.altering  the  rider  one  place,  find  the  resting-point  on  the 
other  side  of  the  zero.  By  interpolation  the  change  of  the 
position  of  the  rider  necessary  to  make  the  resting-point 
coincide  with  the  zero-point  is  deduced.  For  example,  the 


24  MECHANICS. 

zero  is  +o  .  46  ;  with  the  rider  at  4  the  resting-point  is  +0.51; 
with  the  rider  at  5  the  resting-point  is  +  o  .  10.  By  changing 
the  rider  from  4  to  5,  o.ooi  g.  was  added.  To  bring  the 
resting-point  to  the  zero  we  should  have  added  0.05-;- 
(0.51—0.10)  of  o.ooi  g.  or  o.oooi  g.  approximately. 
Hence  the  weight  of  the  body  is  the  weight  on  the  pan  plus 
0.0041  g. 

The  arms  of  the  balance  may  be  unequal.  If  this  be  so, 
the  weight  obtained  above  will  not  be  the  true  weight.  To 
eliminate  this  error  the  body  must  be  changed  to  the  other 
pan  and  another  weighing  made.  If  /  be  the  length  of  the 
left  arm  and  r  that  of  the  right  and  if  u  be  the  counterbal- 
ancing weight  when  the  body  is  in  the  left  pan  and  v  when 
it  is  in  the  right,  while  w  is  the  true  weight  of  the  body  then, 
lw=ru,  lv=rw 


(The  geometric  mean  of  two  very  nearly  equal  quantities 
is  nearly  equal  to  their  arithmetic  mean.)  The  ratio  of  the 
arms  of  the  balance  may  also  be  calculated,  since 


The  buoyancy  of  the  air  on  the  weights  and  on  the  body 
must  be  allowed  for  in  accurate  work.  To  the  apparent 
weight  of  the  body  must  be  added  a  correction  equal  to  the 
weight  of  the  air  displaced  by  the  body  and  from  the  apparent 
weight  must  be  subtracted  the  weight  of  the  air  displaced 
by  the  weights.  In  each  case  the  weight  of  the  air  displaced 
can  be  calculated  if  its  volume  and  density  are  known. 
This  correction  in  any  case  is  very  small.  A  small  per- 
centage error  in  the  correction  will  not  appreciably  affect 
the  calculated  true  weight.  Hence  approximate  values  of 
the  volumes  of  the  body  and  weights  may  be  used.  In 
finding  the  volume  of  the  weights  the  density  of  brass 
weights  may  be  taken  as  8.4.  The  density  of  air  at  o°  and 
760  mm.  may  be  taken  as  .0013,  and  its  density  at  the 
temperature  of  the  laboratory  and  the  pressure  indicated 


ADJUSTMENT  OF  TELESCOPE  AND  SCALE.        25 

by  the  barometer  may  be  calculated  by  the  laws  of  gases. 
Hence  the  temperature  and  barometric  pressure  should  be 
obtained. 

23.  Adjustment  of  Telescope  and  Scale. 

To  adjust  a  telescope  and  scale,  determine  approximately 
the  location  of  the  normal  to  the  mirror,  either  by  finding 
the  image  of  one  eye  or  the  image  of  an  incandescent  lamp 
held  near  the  eye.  Move  the  stand  supporting  the  telescope 
and  scale  until  the  center  of  the  scale  is  about  in  line  with 
the  normal.  Look  along  the  outside  of  the  telescope  at  the 
mirror  and  move  the  scale  up  and  down,  or,  if  this  is  not 
possible,  raise  or  lower  the  stand  until  you  see  the  reflection 
of  the  scale  in  the  mirror.  It  may  be  a  help  to  illuminate 
the  scale  with  an  incandescent  lamp.  Look  through  the 
telescope  pointed  at  the  mirror,  and  change  the  focus  until 
the  scale  is  seen  distinctly.  Remember  that  the  more 
distant  the  object,  the  more  the  eye-piece  must 'be  pushed  in, 
and  that  the  image  of  the  scale  is  at  about  twice  the  distance 
of  the  mirror. 

24.  Time  Signals. 

A  convenient  source  of  time  signals  for  a  laboratory  is  a 
chronometer  which  either  opens  or  closes  a  circuit  containing 
batteries,  sounders,  etc.,  every  second  with  an  omission 
at  the  end  of  each  minute. 

The  individual  second  intervals  indicated  by  a  chro- 
nometer, so  arranged,  are  likely  to  be  somewhat  inaccurate, 
and  therefore,  when  an  accurate  interval  of  one  second  is 
required,  a  second's  pendulum  should  be  used  with  a  plat- 
inum point  making  contact  with  a  drop  of  mercury,  and  thus, 
if  desired,  closing  an  electric  circuit.  Since  it  is  difficult 
to  set  the  mercury  drop  exactly  in  the  center  of  the  path, 
alternate  seconds  are  likely  to  be  too  long.  Therefore,  if 
possible,  a  two  seconds'  interval  should  be  employed, 
alternate  contacts  being  disregarded.  If  these  contacts 
cause  confusion,  a  pendulum  omitting  alternate  contacts 
may  be  used  (see  Ames  and  Bliss,  p.  486). 


I.  TO  MAKE  AND  CALIBRATE  A  SCALE. 

To  illustrate  the  use  of  the  dividing  engine  (described  on 
page  17)  a  short  scale  is  to  be  engraved  in  millimeters  on  a 
strip  of  nickel-plated  steel  and  then  calibrated  by  compari- 
son with  the  average  millimeter  of  a  standard  scale. 

Arrange  the  cogs  of  the  dividing  gear  so  that  each  fifth 
mm.  division  shall  be  longer  than  the  intermediate  divisions 
and  each  tenth  division  still  longer.  Test  this  adjustment 
on  a  rough  test  strip.  Next  clamp  the  strip  to  be  divided 
on  the  platform  of  the  engine  so  that  it  is  parallel  to  the 
screw;  this  can  be  tested  by  observing  the  edge  of  the 
strip  in  the  microscope  as  the  platform  is  advanced  by  the 
screw.  Care  should  be  taken  to  clamp  the  pillar  that  sup- 
ports the  divider  so  that  the  point  of  the  divider  moves 
perpendicular  to  the  length  of  the  scale.  A  scale  of  2 
or  3  cms.  should  then  be  marked  out  on  the  steel  strip 
and  the  temperature  of  the  platform  ascertained  by  a 
thermometer. 

This  scale  is  next  to  be  calibrated.  The  exact  pitch  of  the 
screw  is  first  obtained  in  terms  of  the  mm.  of  the  standard. 
For  this  purpose  a  considerable  length,  e.  g.,  a  decimeter, 
of  the  standard  should  be  measured  on  the  engine.  This 
should  be  done  for  three  different  parts  of  the  screw.  The 
agreement  of  the  three  determinations  will  afford  some 
indication  of  the  uniformity  of  the  screw.  The  scale  should 
then  be  measured  mm.  by  mm.  For  the  first  reading  the 
circular  scale  of  the  screw  may  be  set  to  zero  when  the 
cross-hairs  coincide  with  the  zero  division  of  the  scale  to 
be  measured,  and  thereafter  the  screw  should  be  turned 
always  in  the  same  direction  and  only  arrested  for  a  reading 
of  the  circular  scale  and  vernier  (the  total  number  of  turns 
being  also  noted)  when  the  microscope  shows  that  the  middle 
of  a  division  has  come  to  coincide  with  the  intersection  of 

26 


ERRORS    OF    WEIGHTS.  27 

the  cross-hairs.  As  this  coincidence  approaches,  the  handle 
should  be  turned  slowly,  and  if  turned  too  far  the  reading 
at  that  point  must  be  omitted  altogether.  The  handle 
should  also  be  turned  slowly  when  contact  with  the  detent 
approaches  so  that  the  screw  may  not  be  arrested  with  a 
jerk. 

As  a  check  on  the  work,  the  whole  length  of  the  scale 
should  be  measured. 

In  calculating  the  true  length  of  the  divisions,  allowance 
must  be  made  for  the  temperature  of  the  standard  which 
may  be  taken  as  the  temperature  of  the  platform  of  the 
engine.  The  standard  is  correct  at  the  temperature  marked. 
From  its  coefficient  of  expansion  calculate  the  length  of  its 
mm.  at  the  temperature  of  observation  and  then  deduce  the 
pitch  of  the  screw  at  the  same  temperature.  Then  from  the 
readings  made,  calculate  the  length  of  each  millimeter  of  the 
scale  and,  by  addition,  draw  up  a  table  showing  the  true 
distance  of  each  division  from  the  zero  division. 

Questions. 

1 .  Enumerate  the  possible  sources  of  error  in  the  use  of  the  divid- 
ing engine  for  the  manufacture  of  scales. 

2.  At  what  temperature  would  the  whole  length  of  your  scale  be 
an  exact  number  of  centimeters?      (Table  VII.) 

II.  ERRORS  OF  WEIGHTS. 

Kohlrausch,  §12;  Watson's  Practical  Physics,  §27. 

Weights  by  good  makers  are  usually  so  accurate  that 
errors  in  them  may  for  most  purposes  be  neglected.  But 
when  less  perfect  weights  are  to  be  used  or  when  weighings 
are  to  be  made  with  the  highest  possible  degree  of  accuracy, 
the  errors  in  the  weights  must  be  carefully  ascertained. 

We  shall  suppose  that  a  100  g.  box  of  weights  is  to  be 
tested,  and  that  a  reliable  100  g.  weight  is  supplied  as  a 
standard,  and  that  an  accurate  10  mg.  rider  is  supplied  for 
making  the  weighings.  The  weights  of  the  box  will  be 
denoted  by  100',  50',  20',  20",  10',  and  so  on,  and  the 


28  MECHANICS. 

sum  5'  +  2'  +  2"  + 1'  by  10".  To  find  the  six  unknown  quan- 
tities, 100',  50',  20',  20",  ior,  10",  we  must  make  six  weigh- 
ings and  obtain  six  relations  between  these  quantities. 
Such  a  set  of  weighings  are  indicated  in  the  following  table. 
Each  should  be  performed  by  the  method  of  double-weigh- 
ing described  on  page  24. 

10'  =10"  +a 

20'  =10'  +  10"  +b 

20"  =20'  +c 

50'  =20'  +20"  +10'  +d 
ioc/  =50'  +  20'  +20"  +  10'  +e 
100    =  100'  +/ 

To  solve  these  equations,  substitute  the  value  of  10'  given 
by  the  first  in  the  second;  then  substitute  the  value  of  20' 
given  by  the  second  in  the  third,  and  so  on  to  the  last,  when 
the  value  of  10"  in  terms  of  the  standard  100  and  a,  b,  c,  d, 
e,  f  will  be  obtained.  The  calculation  of  the  other  quantities 
will  then  present  no  difficulty.  To  standardize  the  box 
completely  the  same  process  must  be  applied  to  10',  5',  2',  2" ', 
i',  i",  and  similarly  to  the  smaller  weights. 


III.  VOLUME,    MASS,    AND    DENSITY    OF    A    REGULAR 

SOLID. 

The  mass  of  the  specimen  (a  sphere  or  cylinder)  is  found 
by  weighing  on  a  sensitive  balance  (see  p.  21).  To  eliminate 
the  inequality  of  the  arms  of  the  balance,  the  body  should  be 
weighed  in  both  pans  (p.  24).  The  zero-point  and  resting- 
points  of  the  balance  should  be  found  by  the  method  of 
vibrations  and  the  various  precautions  in  the  use  of  the 
balance  must  be  carefully  observed.  Allowance  should  be 
made  for  air  buoyancy  (p.  24)  and  corrections  should  be 
applied  to  the  weights,  if  the  weights  have  been  corrected  in 
the  preceding  experiment,  or  if  a  table  of  corrections  is 
supplied. 


MOHR-WESTPHAL    SPECIFIC    GRAVITY    BALANCE.  29 

The  dimensions  of  the  specimen  are  measured  by  a 
micrometer  caliper  (p.  14)  or  a  vernier  caliper  (p.  14). 
If  the  body  is  spherical,  ten  measurements  of  the  diameter 
should  be  made  and  the  average  taken;  if  it  is  cylindrical 
ten  measurements  of  the  diameter  and  ten  of  the  length 
should  be  made. 

From  the  mass  and  the  volume,  the  density  (or  mass  per 
c.c.)  is  deduced. 

The  ratio  of  the  arms  of  the  balance  should  also  be  derived 
from  the  results  of  the  double  weighing  (p.  24). 

The  possible  error  of  the  density  determination  should  be 
calculated  as  illustrated  on  p.  6. 

Questions. 

1.  If  the  object  aimed  at  were  merely  the  density  of  the  body, 
which  of  the  above  measurements  should  be  improved  in  precision 
and  to  what  extent  would  it  need  to  be  improved? 

2.  If  the  above  improvement  were  not  possible,  how  much  of  the 
refinement   of   measurement  of  the  other  quantity  might  be  dis- 
carded ? 


IV.  MOHR-WESTPHAL  SPECIFIC  GRAVITY 
BALANCE. 

Kohlrausch,  p.  45;  Stewart  and  Gee,  I,  §92,  III. 

This  is  a  convenient  form  of  hydrostatic  balance  for 
rinding  the  density  of  a  liquid  by  determining  the  buoyancy 
of  the  liquid  on  a  float  hung  from  an  arm  of  the  balance 
and  immersed  in  the  liquid.  Instead  of  weights  riders  are 
used,  the  arm  of  the  balance  from  which  the  float  hangs 
being  graduated  into  ten  divisions.  The  float  is  made  of 
such  a  size  that  when  hanging  in  air  from  the  graduated 
arm  of  the  balance  (which  is  less  massive  than  the  other 
arm)  it  will  just  produce  equilibrium.  Four  riders  of  differ- 
ent mass  are  employed,  each  one  being  ten  times  as  heavy 
as  the  next  smaller.  The  largest  rider  is  of  such  a  size 
that  if  the  float  hanging  from  the  balance  be  immersed  in 
water  at  15°  C.  the  addition  of  the  rider  to  the  hook  at  the 


30  MECHANICS. 

end  of  the  beam  will  restore  equilibrium.  Hence  it  counter- 
balances the  buoyancy  of  the  water  on  the  float.  Thus  it  is 
evident  that  if  the  water  be  replaced  by  a  liquid  of  unknown 
density  at  the  same  temperature  (so  that  the  volume  of  the 
float  is  the  same)  and  if  the  largest  rider  under  the  circum- 
stances produces  equilibrium  when  placed  at  the  sixth 
division,  then  for  equal  volumes,  this  liquid  can  weigh 
only  six-tenths  as  much  as  water,  or  its  density  is  0.6.  A 
second  rider,  one-tenth  as  heavy  as  the  first,  would  evidently 


FIG.  7. 


enable  us  to  carry  the  process  one  decimal  place  farther, 
etc.  For  liquids  of  a  density  exceeding  unity,  another 
rider  equal  to  the  largest  must  be  hung  from  the  end  of  the 
beam,  and  still  a  third  may  be  necessary  for  liquids  of  den- 
sity above  2. 

From  the  above  it  will  be  seen  that  (i)  the  balance  must 
be  adjusted  by  the  leveling  screw  on  the  base  until  the  end 
.of  the  beam  is  opposite  the  stud  in  the  framework  when  the 
float  is  suspended  in  the  air;  (2)  the  beaker  must  always 
be  filled  to  the  same  level,  that  level  being  such  that  when 
the  liquid  is  water  at  15°  C.  the  balance  is  in  equilibrium 
with  the  largest  rider  hanging  above  the  float,  and  (3)  the 
liquid  tested  must  be  at  15°  C. 


MOHR-WESTPHAL    SPECIFIC    GRAVITY    BALANCE.  31 

As  an  exercise  in  the  use  of  this  balance,  find  what  shrink- 
age of  volume  there  is  in  the  solution  of  some  salt  (e.  g., 
common  salt,  ammonium  chloride  or  copper  sulphate) 
in  water  and  find  how  the  shrinkage  varies  with  the  con- 
centration. Solutions  may  be  made  up  by  weighing  out 
very  carefully  on  a  sensitive  balance  (see  p.  21),  0.5  gm., 
i  gm.,  4  gm.,  10  gm.,  etc.,  of  the  salt  and  dissolving  each  in 
a  deciliter  of  water.  When  the  density  of  a  solution  has 
been  found,  the  percentage  contraction  is  calculated  from 
the  sum  of  the  volumes  of  the  constituents  before  mixture 
and  the  volume  of  the  solution  after  mixture;  the  volume 
in  each  case  equals  the  mass  divided  by  the  density.  The 
densities  of  various  salts  are  given  in  Table  VII. 

The  densities  found  and  the  percentages  of  contraction 
should  be  represented  by  curves  with  percentages  of  salt 
as  abscissae.  If  any  determination  of  density  be  largely  in 
error  it  will  be  shown  by  the  curve. 

If  time  permit,  determine  the  density  at  1 5°  of  equivalent 
solutions*  of  several  salts  having  the  same  base,  e.  g., 
NaCl;  1/2  Na2SO4;  NaNO3;  etc.,  and  compare  with  the  densi- 
ties of  similar  solutions  with  a  different  base,  e.  g.,  NH4C1; 
1/2  (NH4)2SO4,  NH4NO3,  etc.  The  difference  in  density 
between  corresponding  salts  should  be  approximately 
constant  (Valson's  Law  of  Moduli).  Find  similarly  the 
difference  in  densities  contributed  by  the  acid  radicals, 
e.  g.,  NaN03  and  NaCl;  NH4N03  and  NH4C1,  etc. 

Questions. 

1 .  What  sources  of  error  may  there  be  in  a  determination  of  density 
by  this  method  ? 

2.  How  might  the  accuracy  of  the  riders  be  tested? 

3 .  How  might  the  accuracy  of  graduation  of  the  beam  be  tested  ? 

4.  What  effect  has  capillarity? 

5.  Explain  the  Law  of  Moduli. f 

*  The  chemical  equivalent  of  a  substance  is  the  atomic  or  molecular  weight 
divided  by  the  valency.  Two  solutions  are  equivalent  if  the  number  of  grams 
of  each  dissolved  in  one  liter  (or  that  proportion)  is  the  same  fraction  of  the 
respective  chemical  equivalent. 

t  Phy.  Chem.,  Ewell,  p.  159. 


32 


MECHANICS. 


V.  DENSITY  BY  VOLUMENOMETER. 

Gray's  Treatise  on  Physics,  I,  §426. 

When  the  density  of  such  substances  as  gunpowder, 
sugar,  starch,  etc.,  is  to  be  determined,  neither  the  method 
of  immersion  in  a  liquid  nor  that  of  the  direct  measure- 
ment of  mass  and  volume  can  be  employed.  The  method 
then  usually  employed  is  that  of  the  volumenometer.  This 
is  a  method  of  immersion  in  air  instead  of  immersion  in 
water,  with  an  application  of  Boyle's  Law 
instead  of  Archimedes'  principle.  The  volume 
of  the  body  is  found  by  placing  it  in  a  glass 
vessel  and  noting  how  much  the  pressure  in 
the  vessel  changes  when  the  air  is  allowed  to 
expand. 

A  gas-washing  bottle  of  about  150  c.c.  capa- 
city, A,  into  which  the  body  is  to  be  introduced, 
is  connected,  by  heavy  pressure  tubing,  with 
an  open,  U-shaped,  mercury  manometer  (see 
Fig.  8).  The  bottle,  A,  is  closed  by  the  stopper 
6,  which  should  be  lubricated  with  rubber 
grease,*  and  forced  into  A  to  a  definite  mark. 
DE  is  raised  until  the  mercury  in  the  burette  BC  is  at  a 
division  B  which  is  carefully  observed.  The  pressure, 
P,  in  A  is  carefully  determined  from  the  difference  in 
mercury  levels  and  the  barometer.  By  the  use  of  a 
rear  mirror,  parallax  may  be  avoided  and  a  small  square 
will  assist  in  reading  a  scale  between  the  two  arms  of  the 
manometer.  The  accuracy  of  the  readings  may  be  increased 
by  using  a  cathetometer  (p.  19).  Lower  DE  until  the  mer- 
cury is  at  a  division  K  and  again  determine  the  pressure,  p. 
Let  the  volume  between  B  and  K  be  v.  Let  V  be  the 
volume  of  A,  and  connecting  tubing,  to  B.  By  Boyle's 
Law: 


FIG.  8. 


*  Equal  parts  pure  rubber  gum,  vaseline,  and  paraffin.  The  two  latter  are 
melted  together  and  the  rubber  is  cut  into  small  pieces  and  dissolved  in  the 
heated  liquid, 


DENSITY   OF  AIR.  33 

Make  at  least  six  determinations  of  P  and  p,  bringing 
the  mercury  each  time  to  the  same  points  B  and  K  which 
should  be  as  far  apart  as  is  convenient.  Calculate  V  from 
the  mean  values. 

Now  introduce  a  carefully  weighed  amount  of  the  assigned 
powder  into  the  bottle,  A  (which  may  be  disconnected  at 
e),  and  insert  the  stopper,  b,  to  its  former  depth.  Again 
determine  the  pressures,  P'  and  pr  when  the  mercury  level 
is  at  B  and  K  respectively.  Repeat  as  before.  If  x  is  the 
unknown  volume  of  the  powder,  the  previous  equation 
becomes 


from  which  x  may  be  calculated.  From  the  volume  and 
mass  of  the  powder  its  density  is  determined. 

If  time  permit,  determine  the  density  also  with  a  specific 
gravity  flask  (pyknometer)  .  Weighings  should  be  made 
of  (i)  bottle  empty;  (2)  bottle  filled  with  a  liquid  of  known 
density  which  is  inert  toward  the  body,  and  (3)  with  a 
known  mass  of  the  body  in  it,  the  rest  of  the  bottle  being 
filled  with  the  liquid.  An  equation  for  density  can  be 
worked  out. 

The  possible  error  in  the  determination  of  the  density 
is  found  by  methods  explained  on  pages  3,8. 

Questions. 

1.  What  sources  of  error  remain  uneliminated? 

2.  With  a  view  to  greater  accuracy  what  suggestions  would  you 
make  as  to  the  most  suitable  magnitudes  for  x  and  v? 

VI.  DENSITY  OF  AIR. 

The  density  of  air  at  atmospheric  pressure,  or  its  mass 
per  cubic  centimeter,  might  be  obtained  by  weighing  a  flask 
containing  air  at  atmospheric  pressure  and  then  re-weigh- 
ing it  after  all  the  air  has  been  removed  by  an  air-pump. 
The  difference  of  weight,  together  with  the  volume  of  the 
flask,  would  give  the  density  of  the  air.  In  practice  the 
3 


34  MECHANICS. 

procedure  has  to  be  modified,  because  it  is  impossible  to 
completely  exhaust  the  flask  of  air.  The  modification  con- 
sists in  finding  the  pressure  of  the  air  remaining  in  the  flask 
and  taking  account  of  it. 

Let  D  be  the  required  density  at  the  room  temperature 
and  pressure,  P.  Let  d  be  the  density  of  the  remaining  air 
when  the  pressure  has  been  reduced  to  p.  Let  the  weight 
of  the  flask  when  filled  with  air  be  W  and  let  w  represent 
its  weight  when  exhausted  to  the  pressure  p. 

W-w  =  V(D-d) 
By  Boyle's  Law 

D  _P         D-d_P-p 

~d=~p    "    ~D~       P 
Therefore 


'P-p        V      P-p 

A  convenient  form  of  flask  is  a  round-bottom  flask  from 
which  part  of  the  neck  has  been  cut  off  and  which  is  closed 
by  a  rubber  stopper  containing  a  glass  tube  with  a  glass 
stop-cock.  The  rubber  stopper  will  hold  tighter  if  lubricated 
with  rubber  grease  *  before  insertion. 

If  the  flask,  as  found,  is  dry,  it  will  be  better  to  postpone 
finding  its  volume  until  the  end  of  the  experiment,  as  the 
operation  requires  it  to  be  filled  with  water.  Moreover, 
of  the  two  weighings  for  finding  the  mass  of  air  removed, 
it  is  better  to  make  the  one  with  the  flask  partly  exhausted 
first,  for  the  weighing  with  the  air  admitted  can  be  made 
immediately  after,  without  handling  the  flask  or  removing  it 
from  the  balance,  a  point  of  some  importance  where  the 
difference  of  weight  to  be  measured  is  so  small.  To  save 
delay  in  weighing  the  flask  after  it  has  been  exhausted,  the 
zero  reading  of  the  fine  balance  used  should  be  obtained 
before  the  flask  is  exhausted.  For  the  method  of  accurate 
weighing,  by  oscillations,  see  page  23. 

*See  note,  p.  32. 


DENSITY    OF   AIR.  35 

A  Bunsen's  aspirator  or  a  Geryk  pump  is  satisfactory 
for  exhausting  the  flask.  The  flask  should  be  connected 
to  the  aspirator  or  pump,  through  a  bottle  for  catching  any 
water  or  mercury.  An  open-tube  manometer  connected 
to  the  tube  that  joins  the  aspirator  or  pump  and  flask  will 
give  the  pressure. 

There  should  be  a  stop-cock  or  a  rubber  pinch-cock  in 
the  connection  between  the  manometer  and  the  pump  or 
aspirator.  When  a  sufficiently  high  exhaustion  has  been 
secured  this  cock  should  be  closed  for  several  minutes  to 
ascertain  if  there  is  any  leakage.  If  not,  both  ends  of  the 
manometer  should  be  read  and  the  stop-cock  of  the  flask 
closed.  Before  removal  of  the  flask,  the  other  cock  should 
be  opened  that  the.  rest  of  the  apparatus  may  fill  with  air. 
If  by  any  chance  a  small  quantity  of  water  should  pass  into 
the  manometer,  allowance  should  be  made  for  it,  the  density 
of  mercury  being  taken  as  13.6. 

The  flask  is  then  weighed  as  quickly  as  possible  on  a 
fine  balance,  the  method  of  vibration  being  used.  It  may 
be  necessary  to  hang  the  flask  by  a  fine  wire  to  the  hook 
which  carries  the  pan.  This  weighing  is  repeated  with  the 
stop-cock  open,  but  with  the  flask  otherwise  undisturbed. 
The  atmospheric  pressure  is  obtained  from  a  reading  of  the 
barometer  (see  p.  21). 

The  volume  of  the  flask  may  be  obtained  by  filling  it 
with  distilled  water  and  weighing  it  on  an  open  balance. 
To  get  the  flask  just  filled  to  the  stop-cock,  the  stopper 
(removed  for  filling  the  flask)  should  be  thrust  in  with  the 
stop-cock  open,  the  stop-cock  should  then  be  closed,  and 
any  water  above  the  stop-cock  should  be  removed.  Of 
course,  the  stop-cock  should  be  replaced  at  its  original 
depth,  which  should  be  marked.  The  density  of  water  at 
different  temperatures  will  be  found  in  Table  V. 

When  the  experiment  is  completed,  place  the  open  flask 
inverted  on  a  frame  to  dry,  so  that  it  may  be  ready  for  the 
next  person  who  uses  it. 

The  density  of  dry  air  may  be  found  in  the  same  way, 


36  MECHANICS. 

the  flask  being  several  times  exhausted  and  refilled  through 
a  drying-tube.  Similarly  the  density  of  any  other  gas,  e.  g., 
carbon  dioxide,  may  be  found  by  filling  the  flask  from  a 
generator.  The  gas  must  be  admitted  to  the  exhausted 
flask  very  slowly  and  the  exhaustion  and  filling  must  be 
repeated  to  insure  the  (almost)  complete  removal  of 
the  air. 

In  reporting,  deduce  from  your  measurement  of  the 
density  of  air  or  gas,  its  density  at  o°  C.  and  760  mm.  by 
using  Boyle's  and  Charles'  Laws.  Find  also  the  possible 
error  of  the  measurement  of  density  (p.  5). 

Questions. 

1 .  Would  the  first  results  be  affected  by  the  presence  of  water  in 
the  flask?     Explain. 

2 .  Should  the  flask  weigh  more  filled  with  dry  air  or  filled  with 
moist  air,  both  at  atmospheric  pressure?     Why? 


VII.  ACCELERATION  OF  GRAVITY  BY  PENDULUM. 

Text-book  of  Physics  (Duff},  §117;  Watson's  Physics,  §§112—114; 
Watson's  Practical  Physics,  §§46-49;  Ames'  General  Physics, 
pp.  74,  91,  135;  Crew's  Physics,  §§85,  86. 

The  acceleration  of  gravity,  g,  is  most  readily  obtained 
from  the  length  and  time  of  vibration  of  a  pendulum.  The 
time  of  vibration  of  an  ideal  simple  pendulum,  i.  e.,  a  heavy 
particle  vibrating  at  the  end  of  a  massless  cord  would  be 

,-     T 

/  being  the  length  of  the  pendulum.  If  the  bob  is  a  ball  so 
large  that  the  mass  of  the  suspending  wire  is  negligible, 
the  above  formula  will  apply  provided  the  radius  of  the 
ball  is  negligible  compared  with  the  length  of  the  pendulum. 
If  these  assumptions  may  not  be  made,  the  pendulum  must 
be  regarded  as  a  physical  pendulum  and  its  moment  of 


ACCELERATION    OF    GRAVITY    BY    PENDULUM.  37 

inertia    about    the    suspension    considered.     Under    these 
circumstances  the  formula 


t  = 


Mgh 


must  be  used,  where  /  is  the  moment  of  inertia  of  the  entire 
pendulum  about  the  knife-edge,  M  is  the  total  mass  and  h 
is  the  distance  from  the  knife-edge  to  the  center  of  gravity 
of  the  whole.  If  the  mass  of  the  suspension  is  negligible 
it  is  only  necessary  to  consider  the  moment  of  inertia 
of  the  ball  about  the  knife-edge.  It  is  easily  shown  that 
the  latter  formula  then  reduces  to  the  formula  for  the 
simple  pendulum,  provided  the  length  of  the  pendulum  is 
taken  as  the  distance  from  the  knife-edge  to  the  center  of 
the  ball  plus  2r2/  $1  where  r  is  the  radius  of  the  ball.  Hence 
to  find  g  there  are  three  quantities,  t,  I,  and  r,  to  be 
measured. 

A  convenient  form  of  pendulum  consists  of  a  spherical 
bob  into  which  screws  a  nipple  through  which  a  fine  wire 
is  passed  and  secured.  To  the  upper  end  of  the  wire  is 
soldered  a  stirrup  of  brass  which  rests  on  a  knife-edge  of 
steel.  A  short  platinum  wire  should  be  soldered  to  the 
lower  side  of  the  bob. 

For  accurately  measuring  the  length  of  the  pendulum  a 
cathetometer  (see  p.  19),  which  should  be  carefully  adjusted, 
may  be  used.  (If  necessary,  the  measurement  of  length  may 
be  postponed  until  the  time  has  been  observed).  The 
horizontal  cross-hair  of  the  cathetometer  is  first  focused 
on  the  knife-edge,  the  fine  screw  being  used  for  the  final 
adjustment  of  the  telescope,  and  the  scale  and  vernier  are 
then  read.  The  telescope  is  then  lowered  and  set  on  either 
the  top  or  bottom  of  the  bob,  whichever  is  the  more  definite. 
These  readings  should  be  repeated  several  times,  beginning 
each  time  with  the  knife-edge.  If  the  adjustments  are 
imperfect,  the  telescope  should  at  least  be  made  exactly 
level  before  each  reading.  The  diameter  of  the  bob  may  be 


38  MECHANICS. 

measured  by  means  of  a  micrometer  or  a  vernier  caliper 
(see  p.  14). 

For  fixing  the  vertical  position  of  the  pendulum,  two 
vertical  pointers  may  be  so  placed  that,  when  the  pendulum 
is  at  rest,  the  pendulum  suspension  and  two  pointers  are 
in  one  plane.  The  eye  of  the  observer  should  always  be 
kept  in  this  plane  in  using  the  first  two  methods.  The  pen- 
dulum is  set  vibrating  in  an  arc  of  3  or  4  cms.  Several 
attempts  may  be  necessary  to  get  the  pendulum  vibrating 
exactly  perpendicular  to  the  knife-edge  with  the  bob  free 
from  rotation. 

The  time  of  vibration  is  most  readily  obtained  with 
precision  when  the  pendulum  is  very  nearly  a  second's 
pendulum,  i.e.,  when  the  period  of  a  complete  vibration  is 
very  nearly  two  seconds.  For  the  determination  of  the 
period  several  methods  are  available.  The  first  and  roughest 
method  given  below  will  serve  for  adjusting  the  pendulum 
to  the  required  length. 

(A)  In    the    first    method    for    determining    the    period, 
time  is  found  by  the  relay    (p.    25)    and    the  number  of 
vibrations  in  three  minutes  is  counted,  fractions  of  a  vibra- 
tion  being    roughly  estimated.     This    is    repeated   several 
times.     Or  a  stop-watch  or  stop-clock  may  be  used,  but  it 
should   be   rated   by   comparison   with    a    chronometer   or 
standard  clock.     The  stop-watch  is  started  as  the  pendulum 
crosses  the  plane  of  observation  and  "one"  is  counted  the 
next   time   the   pendulum   crosses   the   plane   in   the   same 
direction.     The   watch   is   stopped   on   the    5oth  vibration; 
and  the  whole  repeated  five  times.     The  mean  time  divided 
by  50  will  give  a  fair  value  for  the  period. 

(B)  A  second  and  much  more  accurate  method  of  obtain- 
ing the  time  of  vibration  is  the  method  of  coincidences.     This 
consists  in  finding  the  rate  at  which  the  pendulum  gains  or 
loses  as  compared  with  a  standard  clock  or  chronometer. 
It  is  applicable  only  when  the  periods  of  pendulum  and  clock 
or  chronometer  are  nearly  the  same  or  when  one  is  nearly 
an  exact  multiple  of  the  other.     The  method  receives  its 


ACCELERATION    OF    GRAVITY    BY    PENDULUM.  39 

name  from  the  fact  that  what  is  observed  is  the  "coincidence 
interval"  or  the  interval  between  the  moment  when  a 
passage  of  the  pendulum  through  the  vertical  coincides 
with  some  signal  from  the  clock  to  the  next  time  when  such  a 
coincidence  occurs. 

In  a  coincidence  interval,  the  pendulum  must  gain  or 
lose  one  vibration  as  compared  with  the  chronometer  or 
other  time  standard.  If  n  such  coincidence  intervals  occur 
in  T  sec.,  the  number  of  vibrations  of  the  pendulum  during 
this  time  is  (T±ri).  Hence  if  /  is  the  time  of  one  vibration, 


T±n 
and  the  period  of  a  complete  vibration  is 

27 


T±n 

A  convenient  form  of  signal  is  given  by  the  chronometer 
and  relay  described  on  page  25.  It  is  advisable  to  have  the 
coincidence  interval  something  between  30  seconds  and  3 
minutes,  and,  if  necessary,  the  length  of  the  pendulum  should 
be  changed  for  the  purpose. 

After  the  coincidence  interval  has  been  roughly  deter- 
mined by  a  few  observations,  the  following  modification  of 
the  method  will  give  it  much  more  accurately.  Calling  the 
time  of  the  first  coincidence  zero  seconds,  observe  the  sec- 
ond on  which  the  next  coincidence  occurs  and  then  the  next, 
until  four  have  been  observed.  Then,  after  allowing  a 
considerable  number  of  coincidences  to  pass  unnoted,  but 
keeping  note  of  the  time,  observe  the  number  of  the  seconds, 
counted  from  the  original  coincidence,  upon  which  four  more 
successive  coincidences  occur. 

From  the  first  set  of  coincidences,  three  estimates  of  the 
coincidence  interval  will  be  obtained  and  three  others  from 
the  second  set,  the  mean  of  all  giving  an  approximate  esti- 
mate. Then  let  the  time  of  the  first  coincidence  of  the  first 
set  be  subtracted  from  the  time  of  the  first  of  the  second 


40  MECHANICS. 

set,  also  the  time  of  the  second  coincidence  of  the  first  set 
from  that  of  the  second  of  the  second  set,  etc.  These  dif- 
ferences give  four  estimates  of  the  time,  T;  of  some  unknown 
integral  number,  n,  of  coincidence  intervals.  If  the  mean 
of  these  four  estimates  be  divided  by  the  mean  time  of  a 
single  coincidence  interval  as  already  found,  the  quotient 
will  be  n  plus  or  minus  a  small  fraction.  This  fraction  is 
due  to  inaccuracy  in  the  estimates  of  the  coincidence  intervals 
and  should  be  dropped.  The  period  /  of  the  pendulum  may 
now  be  calculated.  The  plus  sign  in  the  denominator  is 
used  if  the  pendulum  is  the  faster. 

The  following  aid  to  the  observation  of  coincidences  is 
suggested.  Keeping  the  eye  constantly  in  the  proper  plane 
for  observation,  make  a  dot  on  a  piece  of  paper  at  each  click 
of  the-  relay.  When  there  appears  to  be  coincidence,  pro- 
long the  dot  into  a  stroke.  To  avoid  recording  every  click, 
a  cross  may  be  used  instead  of  a  dot  for  marking  a  minute, 
and  the  clicks  may  be  passed  unrecorded  until  the  next 
minute,  or  coincidence.  There  may  be  several  successive 
clicks  during  which  there  appear  to  be  coincidences,  in 
which  case  several  successive  strokes  should  be  made  and 
the  mean  taken.  From  these  dots,  strokes,  and  crosses,  the 
times  of  coincidence  may  be  deduced.  Or,  a  dial  indicating 
seconds  may  be  employed,  the  second  when  there  first 
appears  to  be  a  coincidence  being  observed  and  the  second 
when  there  first  appears  to  be  no  coincidence.  Since  the 
clock  cannot  be  observed  immediately,  the  ticks  are  counted 
until  the  clock  is  observed  and  then  subtracted ;  minutes  must 
be  noted  and  recorded  if  they  are  not  recorded  on  the  clock. 

(Q  A  third  method  consists  in  modifying  the  second 
method  so  that  coincidences  of  two  sounds  are  observed. 
The  pendulum  is  made  to  actuate  a  sounder  or  telephone 
each  time  it  passes  through  the  vertical  and  a  coincidence  is 
observed  when  the  sounder  and  relay  strike  together.  A 
block  of  wood  with  a  narrow  trough  filled  full  of  mercury 
is  placed  in  a  mercury  tray  and  is  adjusted  beneath  the 
pendulum  so  that  the  platinum  wire  on  the  under  side  of  the 


COEFFICIENT    OF    FRICTION.  41 

bob  just  touches  the  mercury  when  the  pendulum  is  at  rest, 
and  crosses  the  narrow  trough  at  right  angles  when  the 
pendulum  is  in  motion,.  A  wire  soldered  to  the  knife-edge 
is  connected  in  series  with  several  batteries,  a  sounder  or 
telephone,  and  the  mercury  trough.  The  final  adjustment 
of  the  mercury  trough  is  made  with  the  leveling  screws  of 
the  mercury  tray.  Care  should  be  taken  not  to  spill  the 
mercury. 

From  the  possible  errors  in  the  measurements  of  /  and  t 
deduce  the  possible  errors  in  the  value  found  for  g 
(see  p.  7). 

Questions. 

1.  Does  the  friction  of  the  knife-edges  and  of  the  air  increase  or 
decrease  the  value  of  g? 

2.  Why   should   coincidence   be   observed   exactly,  for   the   plane 
containing  the  position  of  rest? 

3.  What  would  be  the  result  of  increasing  the  arc  of  vibration  to 
10  cm.?      (Table  III.) 

4.  Why  should  the  top  reading  of  the  cathetometer  always  precede 
the  bottom  reading? 

5.  Design,    if    possible,    a    scheme    of    electrical  connections  such 
that  the  sounder  will  only  operate  when  there  is  a  coincidence. 


VIII.  COEFFICIENT  OF  FRICTION. 

Text-book  of  Physics  (Duff),  §§126—130;  Watson's  Physics,  §§96-100; 
Ames1  General  Physics,  p.  118;  Crew's  Physics,  §117;  DanielVs 
Physics,  pp.  176-184. 

The  coefficient  of  friction  of  two  surfaces  is  the  ratio 
of  the  force  of  friction  opposing  the  incipient  or  actual 
relative  motion  to  the  force  pressing  the  two  surfaces  to- 
gether. The  force  requisite  to  start  the  motion  is  greater 
than  that  required  to  sustain  the  motion,  i.  e.,  the  "coeffi- 
cient of  static  friction"  is  greater  than  that  of  "kinetic  fric- 
tion." Moreover,  the  coefficient  of  kinetic  friction  is  not 
quite  constant,  but  varies  somewhat  with  the  speed. 

(A)  The  coefficient  of  static  friction  of  one  surface  on 
another  may  be  found  by  means  of  a  block  of  the  former 
resting  on  a  slide  of  the  latter.  One  end  of  the  slide  is  gently 


42  MECHANICS. 

elevated  by  a  screw  until  the  block  just  fails  to  stand  sta- 
tionary on  the  slide.  The  tangent  of  the  angle  which  the 
slide  then  makes  with  the  horizontal  equals  the  coefficient 
of  static  friction  (see  references).  The  tangent  may  be 
measured  by  some  simple  method,  using  meter-stick,  plumb- 
line  and  level  or  square.  Several  entirely  independent 
adjustments  for  this  angle  and  measurements  of  the  tan- 
gent should  be  made,  the  adjusting  screw  being  each  time 
turned  some  distance  down  so  that  the  influence  of  the  pre- 
vious setting  may  be  avoided.  The  friction  may  vary  some- 
what from  point  to  point,  and  if  so,  different  points  should 
be  chosen  for  the  separate  trials. 

The  accuracy  of  the  determination  of  the  tangent  should 
be  calculated  to  see  whether  the  possible  errors  will  account 
for  the  variations  of  the  coefficient.  Such,  however,  will 
probably  not  be  found  to  be  the  case. 

(B)  The  coefficient  of  kinetic  friction  may  be  determined 
by  the  same  apparatus  if  we  can  find  the  acceleration 
with  which  the  block  moves  down  the  slide  when  the  latter 
is  tilted  beyond  the  angle  of  repose.  For,  if  the  acceleration 
of  the  block  is  a  and  its  mass  m  and  the  angle  of  inclination 
of  the  slide  i,  then  the  component  of  gravity  down  the  slide 
is  mg  sin  i  and  the  pressure  on  the  slide  is  mg  cos  i.  Hence, 
if  JJL  is  the  coefficient  of  friction,  by  Newton's  second  law, 

m  a  —m  g  sin  i  —  ta  m  g  cos  i 

a 

and,  «  =  tan  ^ ;. 

g  cos  i 

This  process  will  give  the  mean  coefficient  of  friction  for  the 
range  of  speeds  through  which  the  block  passes,  but  for  the 
low  speeds  in  question  the  coefficient  does  not  vary  much. 

The  acceleration,  a,  is  found  by  a  method  frequently 
employed  in  physical  measurements.  A  tuning-fork  (fre- 
quency of  50  or  less)  is  fastened  in  a  clamp  attached  to  a 
support  above  the  slide.  A  stylus  of  spring  brass  with  a 
steel  needle  point  is  attached  to  one  prong  and  just  behind 
this  stylus  is  a  second  stationary  stylus  which  is  attached  to 


COEFFICIENT    OF    FRICTION.  43 

the  support.  A  long  and  narrow  glass  plate  is  covered  with 
the  washing  compound  called  "Bon  Ami"  by  transferring, 
with  a  wet  cloth,  a  little  of  the  paste  from  the  cake  to  the  glass, 
and  then  spreading  it  out  in  a  thin  layer.  The  block  is 
then  raised  to  the  top  of  the  slide  and  secured  by  a  trigger. 
The  support  that  carries  the  fork  is  raised  and  lowered  and 
the  fork  is  adjusted  in  the  clamp  until  each  stylus  touches 
the  coated  glass,  making  with  it  an  angle  of  about  45°,  the 
stylus  on  the  fork  being  exactly  in  front  of  the  other  stylus. 

The  frame-work  is  then  lifted  until  neither  stylus  touches 
the  glass.  The  fork  is  set  in  vibration  by  drawing  the  prongs 
together  with  the  fingers  and  releasing  them,  or  by  with- 
drawing a  wooden  wedge,  and  is  then  adjusted  until  its 
stylus  vibrates  an  equal  amount  on  each  side  of  the  other 
(stationary)  stylus.  The  frame-work  is  then  lowered  until 
the  styli  touch  the  glass  and  the  block  is  immediately  re- 
leased by  the  trigger. 

A  wave  line  should  be  obtained  with  a  straight  line  ex- 
actly in  the  center,  the  amplitude  of  the  wave  line  on  each 
side  of  the  straight  line  being  several  millimeters.  Since 
in  any  measurement  the  effect  of  inaccuracies  at  the  ends  is 
less  important  the  greater  the  quantity  measured,  we  meas- 
ure the  distance  passed  over  during  several  vibrations  of 
the  fork.  This  distance  divided  by  the  time  in  which  it 
was  traversed,  i.  e.,  by  the  period  of  the  fork  multiplied  by 
the  number  of  vibrations,  gives  the  average  velocity  of  the 
block  during  this  time.  If  T  be  the  period  of  the  fork  and 
x  the  distance  passed  over  in  n  complete  vibrations  of  the 
fork,  the  average  velocity  is  oc/nT.  Similarly  we  find 
the  average  velocity  for  the  next  n  complete  vibrations. 
The  average  acceleration  will  be  the  difference  between 
these  average  velocities  divided  by  their  separation  in  time 
or  nT;  for  since  each  velocity  is  the  average  we  may  con- 
sider it  as  belonging  to  the  middle  of  the  time  for  which 
it  is  the  average.  From  several  successive  groups  of  n 
vibrations  several  values  of  the  acceleration  are  obtained 
and  the  mean  taken. 


44  MECHANICS. 

It  remains  to  determine  the  period  of  the  fork.  Two 
methods  will  be  described,  (a)  The  fork  is  clamped  be- 
side a  small  electro-magnet  connected  through  a  battery 
with  a  pendulum  which  closes  the.  circuit  every  second  (see 
p.  25).  To  the  armature  of  the  electro-magnet  a  stylus  is 
also  attached.  A  plate  of  glass  covered  with  "  Bon  Ami"  is 
clamped  on  a  movable  block  so  that  each  stylus  rests  upon 
it.  The  electro-magnet  and  fork  may  have  any  relative 
position  which  may  be  convenient,  but  the  styli  should  not 
be  far  apart.  The  fork  is  set  vibrating  and  the  block  with 
the  glass  is  drawn  along,  the  fork  making  a  wave  line  and 
the  other  stylus  a  straight  line  broken  (or  notched)  every 
second.  With  a  square,  lines  are  drawn  at  right  angles  to 
the  glass  through  the  beginnings  of  alternate  second  sig- 
nals and  the  number  of  complete  vibrations,  estimated  to 
tenths,  is  counted  between  the  lines. 

(6)  This  is  known  as  a  stroboscopic  method  and  depends 
upon  the  persistence  of  vision.  The  fork  is  watched  through 

holes  in  a  disk  revolving  at  a  con- 
stant speed.  The  holes  are  equally 
spaced  in  concentric  circles,  the 
number  per  circle  increasing  with 
the  radius.  The  speed  of  the  disk 
is  varied  until  the  fork  appears 
stationary  when  viewed  through 
FIG.  9.  the  holes  of  a  particular  circle.  If 

there  are  m  holes  in  the  circle,  and 

if  the  disk  revolves  n  times  per  second,  the  frequency  of  the 
fork  is  obviously  mn.  By  varying  the  speed  and  using  other 
holes,  additional  determinations  may  be  made.  The  speed 
of  the  disk  is  obtained  by  determining,  with  a  counter,  the 
number  of  revolutions  in  a  given  time. 

(C)  Another  method  of  finding  the  coefficient  of  kinetic 
friction  is  to  make  the  slide  horizontal  and  find  the  force 
required  to  keep  the  block  in  uniform  motion  after  it  has 
been  started.  For  this  purpose  a  braided  cord  is  attached 
to  one  end  of  the  block,  passed  over  a  pulley  at  the  end  of 


COEFFICIENT    OF    FRICTION.  45 

the  slide,  and  attached  to  a  scale  pan,  to  which  weights  are 
added.  In  this  case  the  weight  of  the  pan  and  weights 
must  not  be  taken  as  the  force  acting  on  the  block,  for  some 
force  is  required  to  overcome  the  friction  of  the  pulley. 
The  amount  required  must  be  found  by  a  separate  experi- 
ment. Two  pans  are  attached  to  the  ends  of  the  cord 
hanging  over  the  pulley  and  sufficient  equal  weights  are 
placed  on  the  pans  to  make  the  pressure  on  the  pulley  the 
same  as  in  the  main  experiment  where  the  parts  of  the  cord 
were  at  right  angles.  The  additional  weight  on  one  scale  pan 
requisite  to  keep  the  whole  in  constant  motion  when  started 
is  the  force  needed  to  overcome  the  friction  of  the  pulley, 
and  is,  therefore,  the  correction  required. 

With  this  apparatus  we  may  also  test  whether  the  coeffi- 
cient of  friction  varies  when  weights  are  added  to  the  block. 
The  correction  for  friction  of  the  pulley  does  not  need  to 
be  re-determined  experimentally,  but  may  be  calculated 
from  the  former  determination,  on  the  assumption  that  the 
friction  of  the  pulley  is  proportional  to  the  pressure  on  it. 

The  possible  error  in  the  results  of  the  first  and  last 
methods  is  easily  determined.  The  most,  accurate  way  of 
finding  the  possible  error  in  method  (B)  is  by  means  of 
formulae  deduced  by  the  differential  calculus  (see  p.  6), 
but  a  much  simpler  and  a  sufficiently  accurate  method  is 
the  following:  Note  that  an  overestimate  of  i  will  increase 
the  value  of  fj.  and  the  same  will  be  the  effect  of  an  under- 
estimate of  a.  Hence  the  coefficient  should  be  recalculated 
with  tan  i  and  cos  i  increased  and  a  decreased  by  their 
possible  errors  and  the  change  found  in  the,  coefficient  may 
be  taken  as  the  final  possible  error.  The  possible  error  of  a 
may  be  taken  as  its  mean  deviation  and  the  possible  errors 
of  tan  i  and  cos  i  may  be  deduced  from  the  measurements 
from  which  they  were  obtained. 

Questions. 

1 .  In  the  second  method,  why  is  it  desirable  that  the  straight 
line  be  exactly  in  the  middle  of  the  wave  line? 

2.  In  the  third  method,  what  error  would  be  introduced  if  the  cord 
from  the  block  was  not  exactly  horizontal  ? 


46  MECHANICS. 

IX.  HOOKE'S  LAW  AND  YOUNG'S  MODULUS. 

Text-book  of  Physics  (Duff),  §§168,  171,  173;  Watson's  Physics, 
§§172,  173;  Ames1  General:  Physics,  pp.  144,  145,  153,  154; 
Crew's  Physics,  §§126-129. 

H coke's  Law  states  that,  for  small  strains,  stress  and 
strain  are  proportional.  Young's  Modulus,  E,  is  the  con- 
stant ratio  of  stress  to  strain  for  a  stretching  strain,  the 
stress  being  taken  as  the  force  per  unit  cross  section  and 
the  strain  as  the  stretch  per  unit  of  length,  or,  if  F  is  the 
whole  force,  A  the  area  of  cross  section,  Lthe  whole  length, 
and  /  the  increment  of  length, 


(A)  The  quantity  most  difficult  to  measure  is  /,  the  small 
increase  of  length.  If  a  wire  be  supported  at  one  end  and 
force  applied  to  the  other  end,  there  is  danger  that  the 
support  may  yield  slightly,  and  a  slight  amount  of 
yielding  will  cause  a  proportionally  large  error  in 
the  estimate  of  the  small  increase  in  length.  The 
peculiarity  of  the  first  method  described  below  is 
the  means  adopted  to  eliminate  the  yield  of  the 
support.  The  increase  of  the  length  of  the  wire 
under  experiment  is  found  by  comparison  with 
another  wire  under  constant  stretch  attached  to 
the  same  support  as  the  former  wire.  One  wire 

6        carries  a  scale  and  the  other  a  vernier  opposite 
the  scale.     If  there  be  any  doubt  which  is  vernier 
(see  p.  13)  and  which  is  scale,  comparison  should 
be  made  with  an  ordinary  steel  scale.     The  screws 
FIG.  10.      by  means  of  which  the  wires  are  clamped  to  scale 
and  vernier  should  be  adjusted  until  scale  and 
vernier  tend  to  lie  in  one  plane.     A  light  rubber  band  may 
then  be  slipped  over  scale  and  vernier  to  keep  them  together. 
The  stretch  may  be  produced  by  means  of  lead  weights. 
The   value   of   these   weights   should   be   determined   by   a 


HOOKE'S  LAW  AND  YOUNG'S  MODULUS.  47 

platform  balance.  To  produce  a  suitable  stretch  it  may 
be  advisable  to  add  two  or  more  weights  at  a  time.  We  shall 
suppose  that  two  are  added,  but  the  description  can  readily 
be  modified  to  suit  any  number.  The  greatest  weight 
should  not  be  more  than  half  that  required  to  break  the 
wire.  (A  copper  wire  o.oi  sq.  cm.  section  will  break  at  40 
kgs. ;  brass,  60  kgs. ;  iron,  60  kgs.)  Suppose,  then,  two 
weights  are  added  at  a  time  and  each  stretch  observed. 
When  the  maximum  number  has  been  added  the  weights 
should  be  removed  in  the  same  order,  readings  being  again 
taken  as  they  are  removed.  The  whole  series  of  observa- 
tions should  be  repeated  at  least  three  times.  Such  readings 
should  always  be  arranged  in  tables  having  in  a  line  or 
column  all  the  readings  for  a  particular  pair  of  weights. 
The  length  of  the  wire  may  be  measured  by  means  of  a 
long  beam  compass  and  the  diameter  should  be  measured 
at  least  a  dozen  times  at  different  places  and  in  different 
directions  by  means  of  a  micrometer  caliper  (see  p.  14). 

Before  calculating,  the  dimensions  should  be  expressed  in 
centimeters  and  the  weights  in  dynes.  First  find  the  mean 
value  of  /  for  each  pair  of  weights  when  added  and  when 
removed  and  then  the  value  of  F  -=-  /  for  each  of  these  values 
of  /  and  the  respective  F's.  Find  the  mean  value  of  F  +  l 
and  the  greatest  percentage  deviation  from  the  mean.  This 
will  give  the  percentage  deviation  from  Hooke's  Law  since 
F-r-l  should  be  a  constant,  A  and  L  being  practically  con- 
stant. The  final  value  of  Young's  Modulus  should  be 
stated  in  the  notation  explained  on  page  n. 

The  possible  errors  of  the  different  quantities  measured 
may  be  taken  as  the  mean  deviation  in  each  case.  The  per- 
centage error  of  the  final  value  of  E  will  be,  as  is  readily  seen 
from  the  formula,  the  sum  of  the  percentage  errors  of  F,  L,  I, 
and  twice  the  percentage  error  of  the  radius  (see  p.  6). 

(B)  Young's  Modulus  may  also  be  found  by  means  of 
the  flexure  of  a  bar.  For,  in  bending  (within  limits)  one 
side  of  a  bar  is  stretched  and  the  other  compressed  (nega- 
tively stretched) ,  and  so  Young's  Modulus  is  the  only  constant 


48 


MECHANICS. 


that  need  be  considered.  The  amount  of  bending  might  be 
deduced  from  the  sag  of  the  center  or  end  of  the  bar,  but  a 
much  more  delicate  method  is  the  following  optical  one: 

A  bar  of  rectangular  cross  section  is  laid  on  two  knife- 
edges  and  at  each  end  is  attached  an  approximately  vertical 
mirror  in  mountings  that  admit  of  considerable  adjustment. 
A  vertical  scale,  nearly  in  line  with  the  bar,  is  reflected 
from  the  farther  mirror  into  the  nearer  and  thence  into  a 
telescope  also  nearly  in  line  with  the  bar.  When  a  weight 
is  attached  to  the  center  of  the  bar,  the  bar  is  bent  and 
another  part  of  the  scale  is  reflected  into  the  telescope. 
This  arrangement  serves  to  determine  the  angle  of  bending. 
For  suppose  the  difference  of  the  scale-readings  on  the 
horizontal  cross-hairs  of  the  telescope  be  D  cms.  (Fig.  n) 
and  let  the  distance  between  the  two  mirrors  be  p  and  the 
distance  of  the  scale  from  the  farther  mirror  q,  then,  if  the 
change  of  inclination  of  each  mirror  be  i, 

D 

"f  Q  n    •*  —  _ 
Lctll    l>  —  • 


For  a    consideration  of    figure   11   will  show  that  dl  =  p    tan   zi; 
But  since  i  is  a  small  angle  tan  22  =  2  tan  i  and  tan 


.,  =  q  tan  42. 
^  =  4  tan  i. 


tan 


FIG.  ii. 


From  tan  i,  the  weight  R  in  dynes  applied  at  the  center,  the 
length  of  the  bar  between  the  knife  edges,  /,  the  breadth,  b, 
and  the  thickness,  a,  Young's  Modulus,  E,  is  obtained,  by 
the  equation 


asb  tan  i 


HOOKE'S  LAW  AND  YOUNG'S  MODULUS. 


49 


PROOF. 

Let  the  y  axis  coincide  with  the  radius  from  the  center  of  curvature 
of  the  bar  to  the  center  of  the  bar,  and  let  the  x  axis  be  the  tangent 
to  the  central  axis  at  this  point.  Designate  distances  from  the  elas- 
tic central  axis,  LOM  (Fig.  12),  along  other  radii  by  z.  The  elastic 
central  axis  remains  unchanged  in  length.  The  curvature  at  any 
point  P  on  LOM  is  the  rate  of  change  of  the  directions  of  the  tangent. 
The  angle  the  tangent  line  at  P  makes  with  the  #-axis  is  a  small  one 
and  may  be  taken  as  dy/dx  (which  is  really  the  tangent  of  that 
angle)  .  The  rate  of  change  of  the  direction  of  the  curve  at  the  point 
x,  y,  is  therefore  d2y/dx'2,  which  therefore  equals  the  curvature.  But 
the  curvature  also  equals  i/r,  r  being  the  radius  of  curvature.  Hence 


Now  consider  two  strips  of  the  beam  distant  ±z  from  LOM.  By 
the  bending  these  strips  are  changed  in  length  in  the  proportion  z/r  or 
2  d2y/dx2.  (For,  consider  figure  13;  the  proportional  change  of 
length  is 

G'H'-GH 
GH 


FIG.  12. 


FIG.  13. 


By  definition  of  Young's  Modulus,  if  a  force  F  applied  to  a  rod  of 
cross  section  A  and  length  L  produce  an  extension  /, 

EAl 


F  = 


L  ' 


where  E  is  Young's  Modulus.  The  stress  in  a  strip  of  width  b  and 
thickness  dz  is  obtained  by  putting  bdz  for  A  and  z  d2y/dx2  for  l-i-L, 
which  gives 


Hence  the  moment  about  P  of  the  restoring  force  in  the  strips  ±z  is 


50  MECHANICS. 

The  moment  about  P  of  the  stress  in  the  whole  cross  section  is  the 
integral  with  reference  to  z  of  the  above  expression  for  values  of  z 
from  o  to  \a  or 


For   equilibrium   this   must    equal  the   moment   of    %R   about  P  or 

d2y^  6R    n 
dy      6R    \lx    x2 


'  dx    Ea3b(2      2  j 
At  a  point  of  support 


Hence  by  substitution 

„  Rl2 


4a36  tan  i' 

(By  integrating  again,  the  value  of  y  at  a  point  of  support  or  the 
deflection  of  the  beam  is  obtained.  This  is  left  as  an  exercise  for 
the  student.) 

The  adjustment  of  the  apparatus  is  most  readily  made 
as  follows.  Place  the  telescope  and  scale  nearly  in  the  line 
of  the  mirrors  and,  glancing  above  the  telescope,  set  the 
farther  mirror  so  that  the  nearer  mirror  is  seen  by  reflection 
and  then  the  latter  so  that  the  scale  is  seen.  Then  adjust 
the  eye-piece  of  the  telescope  so  that  the  cross-hairs  are  as 
distinct  as  possible  and  finally  focus  the  telescope  until  the 
scale  is  seen.  The  bar  must  not  be  strained  beyond  the 
limits  of  elasticity.  For  adjustment  of  the  telescope  and 
scale,  see  p.  25.  Equal  weights,  perhaps  100  grams  at  a 
time,  should  be  added,  but  this  process  should  be  stopped 
when  it  is  found  -that  the  scale-reading  no  longer  changes  in 
the  same  proportion  as  the  weights.  Determine  carefully 
by  several  readings,  with  and  without  this  maximum  weight 
attached,  the  change  of  scale-reading.  The  width  and 
thickness  of  the  bar  may  be  measured  by  a  micrometer 
caliper  (see  p.  14),  a  number  of  readings  of  each  at  different 
points  being  made. 

In  calculating,  use  this  weight  for  R  and  the  average  of 
the  changes  of  deflection  for  D  and  take  the  mean  deviation 


RIGIDITY.  51 

as  the  measure  of  the  possible  error  of  D,  a,  and  b.  The 
percentage  possible  error  of  tan  i  is  deduced  from  the  possible 
errors  of  D  and  2p+4q.  The  possible  error  in  the  latter 
term  is  twice  the  possible  error  in  p  plus  four  times  that  in  q. 

(C)  A  simple  optical  method  may  also  be  employed  for 
finding  the  extension  of  a  wire.     In  this  method,  one  side 
of  a  small  bench  carrying  a  vertical  mirror  is  supported  by 
the  end  of  the  wire  and  the  other  by  a  fixed  bracket.     The 
deflection  of  the  mirror  when  weights  are  added  to  the  wire 
is  read  by  a  scale  and  telescope.     The  details  of  the  method 
may  readily  be  worked  out  by  anyone  who  has  followed  the 
preceding  methods. 

(D)  (Searle's  Method.)     The  extension  may  also  be  deter- 
mined from  the  change  of  position  of  a  level  supported  by 
the  two  wires.     The  lowering  of  the  stretched  wire  is  com- 
pensated by  a  micrometer  screw  which  therefore  reads  the 
extension.     For  details,  see  Watson's  Practical  Physics,   §45. 

Questions. 

1.  How  closely  is  it  worth  while  to  measure  the  length  of  the  wire 
in  the  first  method  ? 

2.  Which  of  the  first  two  methods  is  the  more  accurate  and  what 
is  the  chief  weakness  of  the  other  ? 

3.  In  the  second  method  why  is  nothing  said  as  to  the  distance 
of  the  mirrors  beyond  the  knife-edges?     Might  they  be  placed  inside? 

4.  Reduce  your  results  to  tons  and  inches. 


X.  THE  RIGIDITY  (OR  SHEAR -MODULUS). 

Text-book  of  Physics  (Duff},  §§119,  170;  Watson's  Physics,  §171,  174, 
175;  Ames'  General  Physics,  pp.  151—153;  Crew's  Physics,  §§131, 
132;  Duff's  Mechanics,  §§117,  130,  131. 

The  rigidity  of  any  material  is  the  resistance  it  offers 
to  change  of  shape  without  change  of  volume.  It  is  meas- 
ured by  the  ratio  of  the  shearing  stress  to  the  shear  pro- 
duced. In  the  twisting  of  a  wire  or  rod,  within  moderate 
limits,  there  is  no  change  of  volume.  Hence  this  affords 
a  means  of  finding  the  rigidity  of  the  material.  The  con- 
stant or  modulus  of  torsion  of  a  particular  wire  is  the  couple 


52  MECHANICS. 

required  to  twist  one  end  of  unit  length  of  the  wire  through 
unit  angle,  the  other  end  being  kept  fixed.  If  it  be  denoted 
by  r  and  if  the  length  of  the  wire  be  L  the  couple  required  to 
twist  the  wire  through  unit  angle  is  r/L.  If  now  to  the  wire 
be  attached  a  mass  of  moment  of  inertia,  /,  and  the  wire 
and  the  mass  be  set  into  torsional  vibrations,  the  time  of  a 
semi-vibration  is,  by  the  principles  of  Simple  Harmonic 
Motion  (see  references)  , 


If  t,  L  and  /  be  found  r  can  be  deduced.  From  the  modulus 
of  torsion  of  the  particular  wire  the  rigidity  n  of  the  material 
of  which  the  wire  consists  can  be  deduced  ;  for 


2T 


Proof. 

Suppose  unit  length  of  the  wire  to  be  twisted  through  unit  angle. 
The  vibrations  are  due  to  the  restoring  couple  at  the  lower  end  pro- 
duced by  the  twist.  Let  the  cross  section  of  the  end  be  divided  into 
concentric  rings  and  let  the  radius  of  one  ring  be  x  and  its  width  dx; 
its  area  is  z-xxdx.  Relatively  to  the  fixed  end  it  is  displaced  through 
unit  angle.  Hence  the  linear  displacement  (supposed  small)  of  the 
ring  whose  radius  is  x  is  x  times  unit  angle  or  simply  x.  This  is  by 
definition  the  shear  and  hence  the  shearing  stress  is  nx.  This  is 
the  restoring  force  per  unit  area  of  the  cross  section.  Hence  the 
restoring  force  of  the  ring  whose  radius  is  x  is  2xnx2dx.  The  effect 
of  this  force  in  producing  rotation  depends  on  its  moment  about  the 
axis  or  2nnx*dx.  The  moment  of  the  restoring  force  of  the  whole  end 
section  is  the  sum  of  expressions  like  2xnx3dx  for  values  of  x  between 
o  and  r  or  ^nnr*.  This  is  by  definition  the  modulus  of  torsion,  r  and 
gives  us  the  above  equation.  It  should  be  noticed  that  it  is  not  a 
constant  for  the  material  of  the  wire,  but  depends  on  the  dimensions 
of  the  particular  wire. 

The  length,  L,  may  be  measured  by  means  of  a  long 
beam  compass  which  is  afterward  compared  with  a  fixed 
brass  scale.  The  radius,  R,  may  be  measured  by  a  microm- 
eter caliper  (see  p.  14),  measurements  being  made  at  a  great 
many  different  places  and  the  mean  taken. 

The  moment  of  inertia,  /,  of  the  wire  and  attached  mass 


RIGIDITY.  53 

might  be  roughly  obtained  by  calculation,  but  it  is  better 
to  apply  an  experimental  method  that  is  used  in  other  cases. 
This  consists  in  adding  to  the  vibrating  mass,  of  unknown 
moment  of  inertia,  another  mass  of  such  regular  form  that 
its  moment  of  inertia  can  be  accurately  calculated,  and 
finding  the  times  of  vibration  before  (t)  and  after  (T)  adding 
this  mass.  If  the  original  moment  of  inertia  be  /  and  the 
added  moment  of  inertia  i: 


whence 


One  of  the  simplest  forms  of  added  inertia  is  that  of  a  solid 
cylinder  of  circular  cross  section  vibrating  about  an  axis 
through  the  center  of  the  axis  of  the  cylinder  and  at  right 
angles  to  it.  The  vibrating  mass  may  then  be  in  the  form  of 
a  hollow  cylinder  in  which  the  solid  cylinder  may  be  placed. 
If  /  be  the  length  and  r  the  radius  of  the  solid  cylinder  of 
mass  m: 


/I2      r*\ 
=m (  —  +-   . 
\ia      47 


The  quantities  /  and  r  can  be  obtained  with  sufficient  pre- 
cision by  measurement  with  a  steel  scale  divided  to  mm.'s, 
and  m  may  be  found  by  a  platform  balance.  In  the  above 
formula  for  i,  it  is  assumed  that  the  axis  of  rotation  is  per- 
pendicular to  the  axis  of  the  cylinder.  That  this  may  be  so 
the  carrier  must  be  carefully  leveled.  This  may  be  done 
by  supporting  close  under  it  a  rod  that  is  carefully  leveled 
by  a  spirit-level  and  comparing  the  carrier  as  it  swings 
with  the  leveled  rod.  The  end  of  the  vibrating  mass  should 
be  provided  with  an  index,  such  as  a  vertical  needle.  A 
stationary  vertical  wire  is  placed  in  front  of  this  index 
when  the  latter  is  in  the  position  of  rest.  The  body  is  set 


54  MECHANICS. 

vibrating  through  an  angle  of  between  60°  and  90°,  all 
pendulum  vibrations  being  carefully  suppressed. 

The  time  of  vibration  may  be  found  by  much  more  accurate 
methods  than  simply  timing  a  certain  number  of  vibrations. 
The  most  common  methods  for  accurately  timing  vibrations 
are  the  "method  of  coincidences"  and  the  "method  of 
passages."  The  former  is  especially  useful  for  finding  the 
time  of  vibration  of  a  pendulum  whose  half  period  is  approx- 
imately one  second  (Exp.  VII).  The  method  of  passages 
will  be  found  suitable  for  the  present  experiment.  It  con- 
sists in  noting  as  accurately  as  possible  the  time  of  every 
nth  passage  of  the  vibrating  system  through  its  mean  posi- 
tion or  position  of  rest.  The  value  to  be  chosen  for  n  is  a 
matter  of  convenience  when  two  observers  work  together,  one 
counting  the  seconds  and  the  other  noting  the  passages,  or 
when  a  single  observer  has  a  chronometer  in  front  of  him. 
But  a  single  observer  noting  time  by  a  clock  circuit  and 
sounder  should  choose  for  n  an  odd  number  such  that  n 
semi- vibrations  occupy  a  little  more  than  a  minute.  (It  is 
supposed  that  there  is  a  minute  signal,  such  as  the  omission 
of  a  tick;  see  page  25.) 

The  passages  are  observed  as  follows:  After  a  minute 
signal,  the  seconds  are  counted  until  a  passage  occurs,  for 
example,  from  left  to  right.  The  second  and  fraction  of  a 
second  of  this  passage  is  recorded.  The  succeeding  passages 
in  each  direction  are  counted  until  the  minute  signal, 
after  which  the  seconds  are  again  counted  until  the  passage 
occurs  from  right  to  left,  for  which  the  second  and  fraction 
of  a  second  is  recorded.  Obviously,  if  n  has  been  properly 
chosen,  the  passage  just  recorded  is  the  nth.  The  succeeding 
passages  are  counted  until  the  next  minute  §ignal,  after 
which  the  second  and  fraction  of  a  second  of  the  2wth 
vibration  (from  left  to  right)  is  recorded. 

The  following  suggestion  may  aid  in  counting  the  seconds 
and  estimating  fractions  of  a  second.  The  observer  should 
keep  counting  seconds  (not  necessarily  out  loud)  along 
with  the  clock;  when  the  number  of  the  second  is  of  two  or 


RIGIDITY.  55 

more  syllables,  the  accent  should  be  thrown  on  one  syllable 
whose  sound  should  coincide  with  the  tick;  thus,  eleven, 
thirteen,  fourteen,  etc.,  twenty-one,  twenty-to?,  etc.  The 
passage  will  usually  occur  somewhere  between  two  ticks. 
To  estimate  at  what  point  of  time  between  the  two  seconds 
the  passage  takes  place,  the  indications  of  the  eye  may  be 
used  to  reinforce  those  of  the  ear.  Suppose  A  (in  Fig.  14) 
to  be  the  mean  position  of  the  index  on  the  vibrating  body, 
then  if  B  and  C  be  its  positions  at  the  fifth  and  sixth 
ticks,  respectively,  and  if  BA  be  six- 
'*  >  tenths  of  the  distance  BC,  it  is  evident 
«<  tf'Vv'  that  the  true  time  of  passage  is  5.6 
seconds.  With  practice  the  eye  can 
become  very  expert  in  making  such 

judgments,  and,  for  the  purpose  of  attaining  such  skill,  the 
method  should  be  used  from  the  beginning,  although  at  first 
not  much  reliance  can  be  placed  on  the  judgment. 

For  simplicity  of  description  we  shall  suppose  that  n  is  5, 
but  the  proper  substitutions  must  be  made  if  n  has  any 
other  value.  When  the  approximate  time  of  5  vibrations 
has  been  obtained  by  observing  a  few  passages,  all  of  the 
passages  need  not  thereafter  be  observed  in  order  to  ascertain 
when 'each  fifth  passage  is  due,  for  this  can  readily  be  foreseen 
by  adding  to  the  time  of  the  last  observed  passage  the 
known  approximate  time  of  5  vibrations.  Further  assistance 
is  obtained  by  recording  the  time  of  the  o,  loth,  2oth,  etc., 
in  a  second  column,  the  first  column  being  headed  "left 
to  right"  and  the  second  "right  to  left."  In  this  way  such 
a  record  as  the  following  is  obtained: 


M.   S. 

l/U    JX. 

M.   S. 

JX.     LU 

M.   S. 

M. 

S 

(0)   .  .   .  . 

(SO)    ...- 

($)'-••• 

(55).. 

(10) 

(60) 

(15) 

(65) 

(20) 

(70) 

(25) 

(75) 

(30) 

(80) 

(35) 

(85) 

(40) 

(90) 

(45) 

(95) 

56  MECHANICS. 

from  which  the  time  of  vibration  is  calculated  thus : 

M.  s.  M.  s. 

(50)-      (o)  — ....      (55)-      (5)  —  .... 
(60)  — (10)  (65)  —  (15) 

(70)  — (20)  (75)  —  (25) 

(80)  — (30)  (85)  — (35) 

(90)  —  (40)  (95)  —  (45) 

Mean  of  50  vibrations — .  ...  — 

Final  mean  of  50  vibrations  =  .  .  .  .possible  error.  .  .  . 
Final  mean  of  one  vibration  =  .  .  .  .  possible  error .... 
To  find  the  possible  error  in  the  value  found  for  n,  first 
eliminate  r  and  i  from  the  equation  given  above  and  express 
n  in  terms  of  the  quantities  observed  L,  /,  T,  t  R,  m.     (r2/ 4 
is  so  small  compared  with  I2/ 12  that  the  effect  of  the  possible 
error  in  the  former  may  be  neglected.)      T  and  t  come  in 
only  in  the  form  (T2  —  t2)  and  the  possible  error  in  this  may 
be  found  by  methods  stated  on  page  5. 

Questions. 

1.  To  increase  the  accuracy  of  the  result,  which  quantity  would 
have  to  be  measured  more  closely  ? 

2.  What  sources  of  error  are  there  other  than  those  referred  to  in 
the  text? 


XL  VISCOSITY. 

Text-book  of  Physics    (Duff),    §§196-198;    Watson's  Physics,    §161; 
Ames'  General  Physics,  pp.  139,  168. 

A  solid  has  rigidity;  that  is,  it  offers  a  continued  resist- 
ance to  forces  tending  to  change  its  shape.  A  liquid  has  no 
rigidity  and  offers  no  continued  resistance  to  forces  tend- 
ing to  change  its  shape;  that  is,  the  smallest  force  if  given 
time  will  produce  an  unlimited  change  in  the  shape  of  the 
liquid.  But  the  rate  at  which  a  liquid  changes  its  shape 
under  a  given  force  is  not  the  same  for  all  liquids.  Some 
liquids  change  very  slowly  and  are  called  viscous  liquids, 
others  change  rapidly  and  are  called  mobile  liquids.  The 


VISCOSITY.  57 

action  of  both  can  be  stated  in  terms  of  a  property  called 
viscosity. 

The  viscosity  of  a  fluid  may  be  defined  as  the  ratio  of  the 
shearing  stress  in  the  fluid  to  the  rate  of  shear.  From 
this  general  definition  a  simpler  definition  can  be  readily 
deduced.  A  shear  consists  essentially  in  the  sliding  of  layer 
over  layer  and  the  shearing  is  the  force  per  unit  area  re- 
quired to  produce  the  shear.  Hence  we  have  the  following 
equivalent  definition:  "The  coefficient  of  viscosity  is  the 
tangential  force  per  unit  of  area  of  either  of  two  horizontal 
planes  at  unit  distance  apart,  one  of  which  is  fixed  while 
the  other  moves  with  unit  velocity,  the  space  between  the 
two  being  filled  with  the  liquid."  (Maxwell.) 

(A)  The  flow  of  liquid  through  a  capillary  tube  is  essen- 
tially of  the  nature  of  sliding  of  layer  over  layer.  The 
cylindrical  layer  in  immediate  contact  with  the  tube  remains 
fixed  or  at  least  has  no  motion  parallel  to  the  axis  of  the 
tube,  and  the  immediately  adjacent  layer  slides  over  it,  the 
next  layer  slides  over  the  second,  and  so  on  up  to  the  center 
of  the  tube.  (In  a  tube  of  greater  than  capillary  bore  this 
is  not  so,  for  there  are  eddies  in  the  motion.  This  distinc- 
tion is  in  fact  the  best  definition  of  the  term  capillary.) 

Thus,  if  we  measure  the  force  causing  flow  through  the 
tube  and  the  rate  of  flow,  we  shall  be  in  a  position  to  deduce 
the  coefficient  of  viscosity  of  the  fluid.  In  fact,  if  M  be  the 
mass  of  liquid  of  density  d  that  flows  in  time  t,  through  a 
vertical  tube  of  length  /  and  radius  of  bore  r,  and  if  h  be  the 
vertical  distance  from  the  level  of  the  liquid  in  the  reser- 
voir above  the  tube  to  the  lower  end  of  the  tube,  the  coeffi- 
cient of  viscosit  is 


Proof. 


Suppose  all  the  liquid  in  a  capillary  tube  of  length  /  and  radius  r 
to  be  solidified  except  a  tubular  layer  of  mean  radius  x  and  thickness 
dx.  If  there  be  a  difference  of  pressure  p  (per  unit  of  area)  between 
the  two  ends,  the  solid  will  attain  a  steady  velocity  such  that  the 


58  MECHANICS. 

viscous  i 
ends.  I 
ity  that 


viscous  resistance  just  equals  the  whole  difference  of  pressure  on  its 
ends.     Hence  it  follows  from  the  definition  of  the  coefficient  of  viscos- 

}xdx 


pxx 
Hence,  v  =  r   , 

2/73 


If  q  be  the  volume  of  the  core  that  flows  out  per  second, 


.       . 

2/7} 

Suppose  now  another  layer  liquefied.  There  will  follow  a  further 
flow  represented  by  the  same  expression  but  with  a  different  value 
for  x.  Let  the  process  be  continued  until  the  whole  is  liquid,  then 
the  whole  flow  per  second,  Q,  will  be  the  sum  of  all  the  values  of  q 
for  values  of  x  between  o  and  r.  Hence 


If  the  tube  be  vertical  and  the  flow  be  due  to  gravity  only,  instead  of 
p  we  must  put  gdh.  If  M  be  the  mass  of  density  d  that  flows  out  in 
time  t, 


In  the  above  it  was  tacitly  assumed  that  the  liquid  adheres  to  the 
tube  without  any  slip.  If  there  were  any  slip  the  outflow  would  be 
increased  by  it  and  the  above  expression  would  not  hold.  Poiseuille 
and  others  verified  the  above  formula  in  all  cases,  thus  showing  that 
no  slip  occurs.  (A  more  formal  proof  of  the  above  equation  is  given 
in  Tait's  "Properties  of  Matter,"  §317). 

A  piece  of  capillary  tubing  should  be  chosen  whose  bore 
is  as  nearly  as  possible  circular  in  section.  This  can  be 
tested  by  examining  the  ends  under  a  micrometer  micro- 
scope (see  p.  15).  If  the  section  is  found  to  be  nearly  cir- 
cular the  principal  diameters  of  the  bore  should  be  measured. 
This  should,  however,  only  be  regarded  as  a  preliminary 
measurement,  serving  as  a  test  of  the  circularity  of  the  bore 
and  a  check  on  the  following  more  satisfactory  method. 

The  mean  radius  of  the  bore  can  be  best  determined  by 
weighing  the  amount  of  mercury  that  fills  a  measured  length 
of  the  tube.  For  this  purpose  the  tube  should  be  first 
cleaned  by  attaching  it  to  the  end  of  a  rubber  tube,  at 
the  other  end  of  which  is  a  hollow  rubber  ball,  and  thus 
drawing  through  it  and  forcing  out  a  number  of  times  (i) 
chromic  acid;  (2)  distilled  water;  (3)  alcohol,  and  finally 
drying  it  by  sucking  air  through  it.  Then  draw  into  the  tube 


VISCOSITY. 


59 


T 


a  column  of  clean  mercury  and  measure  its  length  as  accu- 
rately as  possible  by  a  comparator  (see  p.  15). 

The  mass  of  the  mercury  should  next  be  ascertained  by 
weighing  it  with  great  care  in  a  sensitive  balance  (for  full 
directions  see  pp.  21-24).  The  mercury  should  not  be 
dropped  directly  on  the  scale  pan,  but  into 
a  watch-glass  or  paper  box  placed  on  the 
scale  pan.  From  these  measurements  and 
the  density  of  mercury  at  the  temperature 
of  observation  (see  Table  VII)  the  diameter 
of  the  bore  is  obtained.  It  may  be  noted 
that  since  it  is  r4  that  is  used  in  the  formula 
for  viscosity  and  r2  that  is  obtained  directly 
from  the  mercury  measurements  of  the  bore 
the  value  of  r  need  not  be  deduced.  The 
length  of  the  tube  may  be  measured  by  the' 
comparator  as  already  described. 

The  tube  is  then  attached  vertically  by  a 
rubber  connection  to  a  funnel  and  the  mass 
of  water  that  flows  through  the  tube  in  a 
given  time  found  by  weighing  a  beaker  (i) 
empty  and  (2)  containing  the  water  that 
has  passed.  The  time  is  obtained  by  ob- 
serving a  clock  ticking  seconds  or  a  chronometer.  It  is 
evident  that  the  greater  the  whole  time  the  less  the  per-, 
centage  error  in  time  due  to  errors  in  observing  the  time 
of  starting  and  stopping,  and  so,  too,  the  greater  the  whole 
mass  the  less  the  percentage  error  in  weight.  Hence  the 
time  and  the  mass  should  be  sufficiently  great  to  make  the 
percentage  errors  in  them  less  than  those  in  /  and  r4.  To 
prevent  evaporation  from  the  beaker  it  should  be  covered 
by  a  sheet  of  paper  pierced  by  a  hole  through  which  the 
tube  passes.  While  the  liquid  is  flowing  the  temperature  of 
the  water  in  the  funnel  should  be  noted. 

The  value  of  h  is  the  mean  of  its  values  at  the  beginning 
and  end  of  the  flow.  These  values  are  best  obtained  by  a 
cathetometer  (p.  19).  For  this  purpose  a  very  simple  form 


FIG.  15. 


60  MECHANICS. 

of  instrument  may  be  used.  A  vertical  scale  is  placed  near 
the  apparatus  for  viscosity  and  the  cathetometer  (a  telescope 
that  may  be  leveled,  movable  along  a  vertical  column  that 
.may  be  made  truly  vertical)  placed  so  that  its  telescope 
(leveled  to  horizontality)  may  be  turned,  so  that  the  inter- 
section of  its  cross-hairs  coincides  alternately  with  the  image 
of  the  water  surface  and  that  of  the  scale.  This  gives  the 
level  of  the  surface  of  the  liquid  on  the  vertical  scale.  The 
level  of  the  lower  end  of  the  vertical  tube  is  obtained  in  the 
same  way,  whence  h  is  obtained. 

The  viscosity  of  alcohol  may  be  measured  by  the  same 
means,  particular  care  being  taken  to  prevent  evaporation. 
The  possible  error  of  the  result  is  readily  calculated  from 
the  possible  errors  of  the  separate  measurements.  The 
possible  error  of  r  is  not  needed,  but  that  of  r4  should  be 
deduced  directly  from  the  determination  of  r2. 

Questions. 

i.  Could  the  radius  be  found  satisfactorily  by  measurements 
with  a  micrometer  microscope?  Explain. 

-2.  What  mass  of  this  liquid  would  flow  through  a  tube  i  mm.  in 
diameter  and  i  meter  long,  under  a  constant  head  of  2  meters  ? 

3.  Two  square  flat  plates  of  20  cm.  edge  are  separated  by  i  mm.  of 
this  liquid.  What  force  would  be  required  to  move  one  with  a 
velocity  of  30  cm.  per  second,  the  other  being  at  rest? 


XII.  SURFACE  TENSION. 

Text-book  of  Physics  (Duff),  §§206—214;  Watson's  Physics,  §§155—160; 
Ames'  General  Physics,  pp.  182-190;  Crew's  Physics,  §§149-160; 
Poynting  and  Thomson,  Properties  of  Matter,  Chap.  XIV. 

The  height  to  which  liquid  rises  or  is  depressed  in  a 
capillary  tube  depends  on  the  surface  tension  of  the  liquid, 
the  angle  of  capillarity,  and  the  radius  of  the  tube.  From 
measurements  of  the  height,  h,  and  radius,  r,  the  surface 
tension  is  deduced  if  the  angle  of  capillarity  is  known,  for 
(see  references) 

rdgh 


T  = 


2  cos  a 


SURFACE    TENSION.  6 1 

d  being  the  density  of  the  liquid  and  g  the  acceleration  of 
gravity.  In  the  case  of  perfectly  pure  distilled  water  the 
angle  of  capillarity  a  or  the  angle  at  which  the  surface  of 
the  liquid  meets  the  glass,  is  zero  and  so  cos  a=  i. 

It  is  important  that  the  capillary  tube  be  quite  clean. 
The  cleaning  should  be  performed  with  chromic  acid  and 
distilled  water.  The  height  of  the  water  in  the  tube  can  be 
measured  in  two  ways.  One  method  is  to  place  a  scale 
etched  on  mirror  glass  behind  the  tube.  The  mean  level 
of  the  meniscus-shaped  surface  of  the  liquid  in  the  tube 
and  the  ordinary  plane  surface  of  the  liquid  in  the  vessel 
should  be  read.  A  preferable  method  is  to  measure  the 
distance  between  the  two  surfaces  with  a  cathetometer 
(see  p.  19). 

To  make  certain  that  the  inner  surface  of  the  tube  is 
wet  by  the  water  and  that  the  angle  of  capillarity  is  zero, 
the  tube  should  be  thrust  deeper  into  the  liquid  and  then 
withdrawn  before  the  levels  of  the  surfaces  are  read.  This 
should  be  repeated  and  the  height  read  several  times, 
different  parts  of  the  scale  being  used,  but  the  part  of  the  tube 
in  which  the  liquid  rises  remaining  the  same.  If  the  motion 
of  the  liquid  in  the  tube  is  sluggish  or  uncertain,  the  tube 
should  be  more  carefully  cleaned.  Finally,  the  point  to 
which  the  liquid  rises  in  the  tube  should  be  marked  on  the 
tube  by  a  sharp  file. 

The  tube  should  then  be  carefully  broken  at  the  point 
marked  and  its  diameter  should  be  carefully  measured  by 
means  of  a  micrometer  microscope  (see  p.  15).  If  the 
section  of  the  bore  is  not  circular,  the  greatest  and  least 
diameters  should  be  carefully  measured  and  the  mean 
taken,  but  if  they  differ  very  much  the  result  will  not  be 
satisfactory. 

The  whole  should  be  repeated  with  as  many  tubes  of 
different  sizes  as  time  will  permit.  The  temperature  at 
which  the  work  is  performed  should  be  stated. 

If  time  permit,  determine  also  the  surface  tension  of  an 
assigned  solution. 


62  MECHANICS. 

Make  an  estimate  of  the  possible  error  for  the  results 
obtained  by  one  of  the  tubes. 

(Apparatus  for  determining  the  surface  tension  at  different 
temperatures  is  described  in  Findlay's  Practical  Physical 
Chemistry,  p.  78;  and  E  well's  Physical  Chemistry,  p.  117.) 

Questions. 

1.  Are  the  errors  of  measurement  sufficient  to  explain  the  dif- 
ferences between  results  with  different  tubes? 

2.  What  other  sources  of  error  may  there  be? 

3.  How  could  the  surface  tension  of  mercury  be  obtained  in  an 
analogous  way? 

4.  How  high  would  this  liquid  rise  in  a  tube  o.  i  mm.  in  diameter? 


HEAT. 

25.  Radiation  Correction  in  Calorimetry. 

Watson's  Practical  Physics,    §82;   Ostwald's  Phys.   Chem.   Meas.,   p. 
124-127;  Poynting  and  Thomson,  Heat,  Chap.  XVI. 

A  body  which  is  above  the  temperature  of  surrounding 
bodies  falls  in  temperature  at  a  rate  that  is  proportional  to 
the  excess  of  its  temperature  above  that  of  its  surround- 
ings. This  is  Newton's  Law  of  Cooling*  If  the  mean 
excess  of  the  body's  temperature  in  any  time  be  known  and 
also  its  rate  of  loss  of  temperature  at  some  particular  ex- 


FIG.   16. 

cess,  its  mean  rate  of  loss  of  temperature  is  readily  deduced, 
and  this  multiplied  by  the  time  for  which  the  mean  is  taken 
will  give  the  whole  loss  of  temperature. 

Consider  the  case  of  the  heating  of  a  vessel  containing 
water  by  the  passage  of  steam  into  the  water.  If  a  curve 
(Fig.  T  6 — from  o  upwards)  showing  the  rise  of  temperature  of 

63 


64  HEAT. 

the  water  be  drawn  and  the  same  continued  after  the  water 
has  reached  its  highest  temperature,  the  latter,  or  straight 
line  part  of  the  curve,  will  give  the  rate  of  loss  of  temperature 
at  the  highest  temperature  attained.  Let  us  denote  this 
rate  by  r  (degrees  per  minute) .  If  the  excess  of  temperature 
when  the  temperature  is  highest  is  t  and  if  the  mean  excess 
during  the  whole  time  of  rise  of  temperature  is  i'  then  the 
mean  rate  of  cooling  was  by  Newton's  Law  rtf \t  and  this 
multiplied  by  the  whole  time  of  rise  of  temperature,  T  (min- 
utes), gives  the  whole  loss  of  temperature.  Hence  the  final 
(highest)  temperature  must  be  corrected  by  addition  of  rTt'/.t. 
If  the  curve  of  rffee  of  temperature  is  a  straight  line,  t'  is  half 
of  t  and  the  correction  is  rT/2.  When  the  curve  is  not 
approximately  a  straight  line  the  whole  time  T  must  be 
divided  into  a  number  of  intervals  (each  perhaps  of  30  sec.) 
and  tf  must  be  obtained  by  averaging  the  excesses  in  these 
intervals. 

When  the  calorimeter  is  cooled  below  the  temperature 
of  the  room  (e.  g.,  by  adding  ice  to  the  water)  the  calor- 
imeter gains  temperature  by  radiation  from  the  surround- 
ings; but  the  above  method  will  still  apply  except  that  we 
shall  have  to  do  with  rates  of  warming  instead  of  rates  of 
cooling  and  the  correction  of  the  final  temperature  will  be 
subtractive. 

If  the  calorimeter  starts  below  the  temperature  of  the 
room  and  is  heated  above  it,  the  correction  must  be  made  in 
two  parts  as  above.  In  this  case  we  must  find  the  initial 
rate  of  warming  (before  the  hot  body  is  placed  in  the  calor- 
imeter) and  also  the  final  rate  of  cooling  (after  the  highest 
temperature  was  attained).  The  correction  will  also  be  in 
two  parts  when  the  calorimeter  starts  above  the  room  tem- 
perature and  ends  below  it. 

If  the  main  rise  of  temperature  is  closely  represented  by 
a  straight  line,  it  is  easily  shown*  that  the  correction  amounts 
to  the  algebraic  average  of  the  initial  and  final  rates  multi- 
plied by  the  whole  time  that  the  calorimeter  is  heating  or 

*  Ewell's  Physical  Chemistry,  p.  84. 


THE  BECKMANN  THERMOMETER.  65 

cooling.  In  fact,  if  the  water  is  Tl  minutes  below  the 
temperature  of  the  room  and  T2  minutes  above  the  room 
temperature,  the  radiation  correction  is  (T2r2  —  T1r1)/2  and 
this  differs  from  (Ti  +T2)(r1  +r2)/2  by  (Tlr2  —  T2rl)/ 2,  which 
is  zero,  since  under  these  circumstances  the  rate  is  propor- 
tional to  the  time  that  the  water  is  above  or  below  the  room 
temperature.  This  fact  is  particularly  useful  in  cases  where 
the  surrounding  temperature  is  indefinite  (Exp.  XXVII,  for 
example) . 

In  every  calorimetry  experiment  where  the  temperature 
changes,  this  radiation  correction  must  be  applied,  and  there- 
fore the  initial  and  final  rates  of  change  of  temperature  must 
be  determined.  The  rate  may  usually  be  found  with  sufficient 
accuracy  by  reading  the  temperature  every  minute  for  five 
minutes.  In  very  accurate  work,  more  careful  methods 
must  be  applied. 

26.  The  Beckmann  Thermometer. 

Watson's  Practical  Physics,  §102;  Findlay,  Practical  Physical  Chem- 
istry, pp.  114—117;  Ostwald,  Phys,  Chem.  Meas.,  pp.  119—120. 

The  Beckmann  thermometer  is  used  for  determining 
changes  in  temperature.  The  bulb  is  large  and  the  stem  is 
small  so  that  a  small  change  of  temperature  is  shown  by  a 
large  change  in  reading.  The  amount  of  mercury  may  be 
varied,  and  the  temperature  corresponding  to  a  particular 
reading  will  vary  with  the  amount  of  mercury  in  the  bulb 
and  stem.  There  is  a  reservoir  at  the  end  of  the  stem  into 
which  surplus  mercury  may  be  driven  by  warming  the  bulb. 
A  gentle  jar  will  detach  the  mercury  in  this  reservoir  when 
sufficient  has  been  expelled.  If  one  desires  to  study  high 
temperature  changes,  the  bulb  is  warmed  until  the  thread 
of  mercury  extends  to  the  reservoir,  when  the  mercury  in 
the  reservoir  is  joined  to  it.  The  bulb  is  then  allowed  to 
cool  until  sufficient  mercury  has  been  drawn  over,  when  the 
thread  is  detached  from  the  mercury  in  the  reservoir  by  a 
gentle  jar.  Several  trials  are  often  necessary  before  the 
5 


66  HEAT. 

proper  amount  of  mercury  is  secured.  In  an  improved  type 
of  Beckmann  thermometer,  two  reservoirs  are  provided,  and 
the  first  has  a  scale  which  tells  the  amount  of  mercury 
required  in  that  reservoir  for  different  ranges  of  temperature. 
Beckmann  thermometers  are  delicate  and  expensive  and 
must  be  handled  with  the  greatest  care. 


XIII.  THERMOMETER  TESTING. 

Watson's  Practical  Physics,  §§59-69;  Edser,  Heat,  pp.  23-36;  Text- 
book of  Physics  (Duff),  pp.  189,  190;  Watson* s  Physics,  §§177- 
182;  Ames'  General  Physics,  pp.  220-224;  Crew's  General  Physics, 
§§249-252. 

The  readings  of  a  thermometer  gradually  change  for  a 
long  time  after  the  thermometer  has  been  filled.  The  cause 
of  this  is  the  gradual  recovery  of  the  bulb  from  the  effect 
of  the  very  great  heating  to  which  the  glass  was  subjected 
when  the  thermometer  was  made.  The  shrinkage  is  rapid 
at  first  and  slower  afterward,  but  may  continue  for  years. 
Hence  the  necessity  for  re-determining,  from  time  to  time, 
the  so-called  "fixed  points"  of  a  thermometer,  namely,  the 
reading  in  melting  ice,  and  that  in  steam  at  standard, pres- 
sure. When  the  thermometer  is  first  graduated  it  is  usually 
done  by  determining  the  fixed  points  and  dividing  the 
distance  between  them  into  100  equal  parts  laid  off  on  the 
stem.  This  assumes  that  the  bore  is  uniform  or  that,  by 
calibration  of  the  bore,  the  variations  of  the  bore  are  deter- 
mined and  allowed  for  in  a  table  of  corrections  to  be  applied 
to  the  readings  of  the  thermometer  in  order  to  obtain  the 
true  temperature.  Usually  the  variations  of  the  bore  are 
too  small  to  have  any  appreciable  effect  except  in  cases 
where  extreme  accuracy  is  aimed  at.  Nevertheless,  every 
thermometer  needs  to  be  carefully  examined  in  this  regard. 
Let  us  suppose  that  on  the  scale  laid  off  on  the  stem  the 
true  readings  in  ice  and  steam  have  been  obtained  and  for 
the  moment  let  us  suppose  that  the  bore  is  quite  uniform. 
To  see  how  to  make  corrections  for  other  points  on 
the  scale  we  must  consider  the  elementary  definition  of 
temperature. 

Temperature  on  the  mercury  scale  is  defined  by  the 
expansion  of  mercury  (relatively  to  glass).  Let  f]00  be 


68  HEAT. 

the  volume  of  a  mass  of  mercury  at  the  temperature  of 
steam  under  a  pressure  of  76  cm.  and  let  VQ  be  its  volume 
at  the  temperature  of  melting  ice.  The  degree  is  defined 
as  the  rise  of  temperature  that  would  produce  an  expansion 
of  (^100  —  ^o)/IOO>  and  T°  above  zero  is,  therefore,  the  rise  of 
temperature  that  will  produce  an  expansion  of  T(v1Q<)  —  v0)  / 1  oo. 
Hence  if  at  T°  the  volume  of  the  mercury  be  v, 


••v—  vf 


IOO 

.'.  T '  =  —        °— 100. 

This  definition  depends  only  on  the  expansion  of  mercury 
and  the  expansion  of  the  particular  glass  used  and  is  other- 
wise independent  of  the  size  and  shape  of  the  thermometer. 
Now  regard  the  thermometer  tube  under  test  as  simply  a 
graduated  cylinder  of  constant  cross  section  containing 
mercury.  Let  the  height  of  the  mercury  as  read  on  the 
scale  when  the  thermometer  is  in  melting  ice  be  a;  when  it  is  a 
steam  at  76  cm.  let  it  be  b,  and  when  it  is  at  the  temperature 
T,  let  it  be  t.  Then 

v—  v0       t—a 
Hence 


J.  \V  \MJ  , 

b—  a 

where  T  is  the  true  temperature  when  the  reading  of  the 
thermometer  is  t.  By  this  equation  values  of  T  for  values 
of  t  for  every  five  degrees  should  be  calculated.  Having 
thus  drawn  up  a  table  of  true  temperatures  we  subtract  the 
scale-reading  from  the  true  temperature  and  thus  get  a 
correction  (positive  or  negative),  which  added  to  the  scale- 
reading  gives  the  true  temperature. 

This  is  on  the  assumption  that  the  bore  is  sensibly  uni- 
form.    The  only  quite  satisfactory  method  of  testing  this 


THERMOMETER    TESTING.  69 

is  to  calibrate  the  bore  by  measuring  the  length  of  a  short 
thread  of  mercury  at  different  positions  in  the  tube.  This 
process  requires  considerable  time  and  the  following  will 
usually  suffice:  Two  thermometers  for  which  tables  of 
true  readings  have  been  drawn  up  as  above,  are  compared 
at  regular  intervals  (say  every  five  degrees)  between  zero 
and  100°  by  being  used  simultaneously  to  measure  the  tem- 
perature of  a  body.  If,  after  corrections,  the  readings  of 
the  thermometers  are  not  sensibly  different,  this  shows  that 
the  bores  of  both  must  be  practically  uniform.  If  they  do 
differ  appreciably,  then  the  bore  of  one  or  both  must  be 
variable.  If  they  be  compared  with  a  third  thermometer, 
the  one  with  the  variable  bore  will  be  detected  and  it  must 
be  then  calibrated. 

Testing  Zero-point. — A  calorimeter  consisting  of  a  small 
copper  vessel  inside  of  a  larger  is  suitable  for  holding  the  ice. 
Both  vessels  should  be  washed  in  ordinary  tap  water. 
The  space  between  the  two  vessels  should  be  filled  with 
cracked  ice,  and  the  inner  vessel  filled  with  cracked  ice  and 
then  distilled  water  poured  in  until  the  vessel  is  filled  to  the 
brim.  The  thermometer  having  been  washed  clean,  is 
inserted  in  the  inner  vessel,  just  sufficient  of  the  stem  being 
exposed  to  admit  of  the  zero  being  observed.  When  the 
reading  has  fallen  to  i°  the  reading  should  be  observed 
every  minute  until  it  is  stationary  for  five  minutes.  This 
stationary  temperature,  read  to  o .  i  degree,  is  the  true  zero 
point,  or  a  in  the  above  equation. 

Sources  of  Error. 

(1)  Impurity  in  the  ice  or  water. 

(2)  The  presence  of  water  above  o°  near  the  bulb  of  the 
thermometer. 

Testing  Boiling-point. — The  form  of  boiler  used  for  this 
test  consists  of  a  vessel  for  boiling  water  surmounted  by  a 
tube  up  which  the  steam  passes,  this  tube  being  enclosed  in 
another  down  which  the  steam  passes  to  an  exit  tube  and  a 
pressure  gauge  (see  Fig.  17).  Half  fill  the  lower  part  of  the 
vessel  with  water.  Push  the  thermometer  to  be  tested 


HEAT. 


] 


through  a  cork  in  the  top  until  the  boiling-point  is  only  a 
degree  or  two  above  the  cork,  but  take  care  that  the 
bulb  of  the  thermometer  does  not  reach  down  to  the 
water.  Apply  heat,  adjusting  it  carefully  as  boiling  begins, 
so  that  the  pressure  inside,  as  indicated  by  the  pressure 

gauge,  shall  not  materially  ex- 
ceed atmospheric  pressure. 
Some  excess  is,  of  course  neces- 
sary, if  there  is  to  be  a  free 
flow  of  steam.  What  excess  is 
permissible  may  be  deduced  from 
the  consideration  that  a  rise  of 
pressure  of  i  cm.  (of  mercury 
column)  corresponds  to  a  rise  of 
boiling-point  of  0.373°  (see  Table 
XIII).  If  water  is  used  in  the 
pressure  gauge,  a  pressure  of  i 
cm.  of  water  column  would  cor- 
respond to  only  0.03°  rise  of 
steam  temperature.  If  the  ther- 
mometer be  graduated  to  degrees 
only,  an  error  of  0.03°  in  finding 
the  boiling-point  is  negligible. 

Read  the  barometer  and  reduce 
the  height  to  zero  degrees  (p.  21). 

To  the  boiling-point  thus 
found  a  correction  must  be  ap- 
plied, for  the  difference  between  the  atmospheric  pressure 
at  the  time  and  that  of  a  standard  atmosphere  (76  cm.  of 
mercury).  Find  from  Table  XIV  the  true  temperature  of 
the  steam  at  this  pressure,  and  the  difference  between  the 
boiling-point  observed  on  the  thermometer  and  this  tem- 
perature. Since  this  temperature  is  always  within  a  few 
degrees  of  100°,  the  thermometer  will  have  practically  the 
same  error  at  100°.  Therefore  b  in  the  above  equation 
may  be  taken  as  100  plus  or  minus  the  difference  between 
the  observed  boiling-point  and  the  true  boiling  temperature. 


FIG. 


TEMPERATURE    COEFFICIENT    OF    EXPANSION.  71 

Comparison  of  Two  Thermometers. — The  most  satisfac- 
tory method  is  to  immerse  the  thermometers  in  steam  above 
water  boiling  under  a  pressure  that  can  be  regulated.  A 
simple  means  that  is  sufficient  if  the  thermometers  are  of 
the  same  length  and  graduated  to  degrees  only,  is  to  use 
the  thermometers  simultaneously  to  find  the  temperature  of 
a  block  of  good  conducting  material  (copper  or  brass)  im- 
mersed in  a  vessel  of  water  the  temperature  of  which  can 
be  gradnally  raised  by  a  burner.  The  thermometers  should 
be  thrust  in  holes  close  together  in  the  block  and  before 
each  reading  the  burner  should  be  removed  and  the  water 
well  stirred  for  a  minute  so  that  the  temperature  of  the 
block  shall  become  uniform. 

Questions. 

1.  Which  should  be  determined    first,   boiling-point   or  freezing- 
point,  and  why? 

2 .  How  much  error  might  there  be  in  determining  the  boiling-point 
if  only  the  bulb  were  immersed  in  the  steam? 

3.  Why  is  there  no  need  to  take  account  of  barometric  pressure 
in  finding  the  zero-point? 


XIV.  TEMPERATURE  COEFFICIENT  OF   EXPANSION. 

Edser,  Heat,  pp.  39-61;  Text-book  of  Physics  (Duff},  pp.  194-197; 
Watson's  Physics,  §§184,  185;  Ames'  General  Physics,  pp.  229— 
233;  Crew's  General  Physics,  §§263-265. 

For  measuring  the  thermal  expansion  of  a  body,  choice 
may  usually  be  made  from  a  variety  of  methods.  The  par- 
ticular method  chosen  will  depend  on  the  form  of  the  speci- 
men. The  expansion  of  a  metal  rod  may  be  measured  by 
means  of  a  spherometer  or  by  means  of  two  reading  micro- 
scopes focused  on  definite  marks  near  the  ends  of  the  speci- 
men. The  expansion  of  a  wire  is  best  measured  by  an  opti- 
cal lever  method.  The  expansion  of  a  solid  of  irregular 
form  can  be  found  by  a  hydrostatic  method,  namely,  by 
weighing  it  in  a  liquid  at  different  temperatures,  it  being 
supposed  that  the  density  of  the  liquid  at  different  tempera- 
tures is  known. 


HEAT. 


(A)  Expansion  of  a  Metal  Rod.- — The  rod  is  supported 
at  the  lower  end  on  a  firm  point  and  is  heated  by  being  en- 
closed in  a  tube  through  which  steam  is  passed  from  a 
boiler.  A  spherometer  (see  p.  16)  is  so  supported  that  the 
end  of  the  screw  can  be  brought  down  on  the  flat  end  of  the 
rod.  The  spherometer  is  supported  in  the 
hole,  slot,  and  plane  method,  so  that  its 
position  is  definite  and  not  liable  to  be  dis- 
turbed by  thermal  expansion  of  the  sup- 
porting surface. 

The  rod  is  first  measured  by  means  of  an 
ordinary  meter  scale  divided  to  mms.  It  is 
then  accurately  placed  in  position  in  the 
heating  tube,  the  end  of  the  rod  projecting 
through  corks.  Through  the  cork  at  the 
upper  end  should  also  pass  a  glass  tube  for 
the  entry  of  the  steam,  while  a  similar  tube 
at  the  lower  end  serves  to  drain  off  the  water. 
At  least  six  readings  of  the  spherometer 
scales  should  be  made  at  the  room  tempera- 
ture. Then  pass  steam  into  the  jacket 
about  the  rod.  Every  few  minutes  read  the 
temperature  of  the  interior  as  given  by  two 
thermometers  at  different  heights  and  read 
the  spherometer.  When  the  temperature 
has  become  constant,  make  at  least  six  read- 
ings of  the  spherometer  and  several  readings 
of  the  thermometer.  Always  estimate  tenths 
of  the  smallest  division.  From  the  differ- 
ence in  spherometer  readings,  the  length,  and  the  change 
in  temperature,  calculate  the  coefficient  of  expansion. 

(B)  Expansion  of  a  Wire. — For  this  an  optical  lever 
method  is  most  suitable.  A  mechanical  lever  or  system  of 
levers  is  sometimes  employed  for  magnifying  small  mo- 
tions. A  ray  of  light  reflected  from  a  mirror  that  is  tilted 
by  the  expansion  serves'  the  purpose  of  a  long  index  arm 
much  better,  inasmuch  as  it  has  no  weight  itself  and  may 


FIG.  18. 


TEMPERATURE    COEFFICIENT    OF    EXPANSION.  73 

be  taken  as  long  as  we  wish.     The  wire  is  hung  vertically, 
the  lower  end  being  solidly  clamped,   and  the  upper  end 
carrying  a  sleeve  on  which  rests  one  leg  of  a  small  three- 
legged  bench,  on  which  a  mirror  is  mounted.     The  other 
two  legs  rest  on  a  fixed  bracket.     The  wire  is  enclosed  by  a 
tube  through  which  a  current  of  steam  is  passed  from  a 
boiler  and  into  which  two  thermometers  are  thrust  to  read 
the  temperature.     A  drainage  tube  at  the  lower  end  allows 
the  escape  of  water.     The  wire  is  prolonged  above  the  mir- 
ror and  is  attached  to  a  spring  by  which  the  wire  is  kept 
stretched.       The    image  in  the 
mirror    of    a    vertical    scale    is          _  __  _____  ----  T 

observed  by  a  reading  telescope  r"~~  ----  -  ------ 

(see    p.    25    for    adjustments),     ^fr-lL^ 
and  the  change  of  reading,  d, 
on    the    horizontal    cross-hairs 

of  the  telescope,  produced  by  the  expansion  is  noted.  Let 
the  length  of  the  wire  between  the  clamp  and  the  support 
of  the  bench  be  /,  and  let  the  length  of  the  bench  between  the 
point  of  the  movable  leg  and  the  line  of  the  other  legs  be  a. 
Let  the  distance  of  the  scale  from  the  mirror  be  L,  and  the 
change  of  temperatures  be  (t2  —  /J  .  Then,  remembering  that 
a  ray  of  light  reflected  from  a  mirror  turns  through  twice  the 
angle  that  the  mirror  turns  through,  it  is  easily  seen  from 
the  figure  that  the  expansion  is  ad/2L  and  the  coefficient 
of  expansion  is 

ad 


The  most  difficult  quantity  to  determine  with  a  high  degree 
of  precision  is  a.  It  may  be  measured  by  means  of  a  mi- 
crometer microscope  or  a  dividing  engine  (see  p.  17).  A 
simpler  and  more  accurate  method  is  to  place  the  optical 
lever  so  that  the  movable  leg  is  on  the  (vertical)  screw  of 
a  micrometer  caliper  (p.  14),  while  the  other  legs  are  on 
a  fixed  support  and  then  focus  the  telescope  and  scale  on 
the  mirror.  When  the  screw  is  turned  the  movable  leg 


74  HEAT. 

is  raised  a  known  amount.  From  this,  the  distance  between 
the  mirror  and  the  scale,  and  the  scale-readings,  a  is  deduced. 
This  calibration  may  be  avoided  by  placing  two  legs 
of  the  mirror  bench  upon  the  collar  attached  to  the  wire 
and  resting  the  third  leg  directly  upon  the  micrometer  screw. 
The  extension  may  also  be  measured  by  Searle's  combina- 
tion of  level  and  micrometer  screw  (see  end  of  Exp.  IX). 

XV.  COEFFICIENT    OF    APPARENT    EXPANSION    OF    A 

LIQUID. 

Edser,  Heat,  pp.  64-71;  Text-book  of  Physics  (Duff),  pp.  198-201; 
Watson's  Physics,  pp.  211-213;  -Awes'  General  Physics,  pp.  233- 
235;  Crew's  General  Physics,  §§266,  267. 

The  object  of  this  experiment  is  to  determine  the  coeffi- 
cient of  apparent  expansion  of  some  salt  solution  with  refer- 
ence to  glass.  A  vessel  holds  M  grams  of  liquid  at  t°  and 
m  grams  at  a  higher  temperature,  t'°.  Let  V  be  the  volume 
of  the  vessel  at  the  lower  temperature.  Since  we  are 
considering  the  apparent  expansion,  i.  e.,  the  expansion 
with  reference  to  the  vessel,  we  may  consider  V  to  be  also 
the  volume  of  the  vessel  at  the  higher  temperature.  The 
volume  of  i  gram  at  t°  is  therefore  V/M  and  at  *'°,  V/m. 
The  increase  in  volume  is 


m     M        \  Mm 
The  coefficient  of  apparent  expansion,  e,  is  this  apparent 
expansion   divided   by  the   original   volume   V/M  and  the 
range  of  temperature  (tf  —  t)  or  . 

M-m 


m(t'-t)' 

A  glass  bulb  with  a  re-curved  capillary  stem  is  used. 
To  fill  the  bulb  with  a  liquid,  warm  it  with  the  hand  or  by 
playing  a  flame  about  some  distance  beneath  it.  Remove  it 
from  the  source  of  heat  and  plunge  the  end  of  the  stem  into 
the  liquid.  As  the  air  in  the  bulb  cools  liquid  will  be  drawn  in. 


COEFFICIENT    OF   APPARENT    EXPANSION    OF   A    LIQUID.       75 

To  expel  liquid,  warm  the  bulb  gently,  keeping  it  so  turned 
that  the  stem  is  filled  with  the  liquid ;  when  the  liquid  ceases 
to  come  out,  invert  it  so  that  the  stem  is  highest,  and  allow 
it  to  partially  cool.  Repeat  until  all  the  liquid  is  expelled. 

Clean  the  bulb  by  drawing  in  a  little  distilled  water,  or, 
if  the  interior  be  foul,  first  use  chromic  acid.  Finally  rinse 
the  interior  with  alcohol.  Remove  the  alcohol  and  dry  the 
interior,  if  necessary  playing  a  flame  about  some  distance 
beneath. 

To  determine  the  density  of  the  (cold)  salt  solution, 
thoroughly  cleanse  a  tall  measuring  glass  and  a  suitable 
hydrometer  (variable  immersion).  Pour  enough  of  the 
salt  solution  into  the  measuring  glass  to  float  the  hydrometer, 
read  the  density,  and  pour  the  solution  back  into  the  bottle. 

Weigh  the  bulb  very  carefully  on  a  sensitive  balance 
(see  pp.  21-25).  Support  the  bulb  in  a  clamp  stand,  clasping 
the  stem  between  half  corks.  Fill  a  small  beaker  with  the 
salt  solution  and  support  it  so  that  the  end  of  the  stem  dips 
into  the  solution.  Warm  the  bulb,  playing  a  Bunsen  flame 
beneath.  Never  allow  the  flame  for  an  instant  to  remain 
stationary  beneath  the  bulb,  and  until  the  bulb  contains  con- 
siderable warm  liquid,  do  not  allow  the  flame  to  touch  the  bulb, 
and  then  only  where  there  is  liquid.  Alternately  warm  the 
bulb  and  allow  it  to  cool  a  little  until  the  bulb  is  filled. 
When  it  is  partly  full1  it  may  be  best  to  gently  boil  the  liquid 
in  the  bulb.  When  the  bulb  is  almost  full  the  liquid  can  be 
made  to  expand  to  fill  the  entire  stem.  Then  allow  it  to 
cool  completely  while  it  draws  over  liquid  from  the  beaker. 

When  the  bulb  is  cooled  to  the  temperature  of  the  room, 
support  it  in  a  copper  vessel  in  which  water  is  kept  at  a 
constant  temperature,  a  few  degrees  warmer  than  the  room. 
When  the  temperature  has  been  kept  constant  for  five 
minutes  (by  the  addition  of  small  amounts  of  hot  or  cold 
water,  if  necessary)  and  has  been  frequently  stirred,  read 
the  temperature  (as  always  estimating  tenths).  Remove 
any  liquid  adhering  to  the  end  of  the  stem,  remove  the  bulb 
from  the  bath,  dry  the  exterior,  and  weigh.  Handle  the 


76  HEAT. 

bulb  carefully  with  a  cloth  about  it  so  that  no  liquid  may 
be  expelled.  Weigh  a  small,  clean,  dry  beaker.  Support 
the  bulb  again  in  the  copper  bath  with  the  beaker  beneath 
the  end  of  the  stem,  to  catch  any  liquid  expelled.  Heat  the 
water  in  the  bath  to  boiling.  When  the  temperature  has 
been  constant  for  live  minutes,  read  the  temperature,  catch 
on  the  side  of  the  small  beaker  any  liquid  adhering  to  the 
end  of  the  stem,  remove  the  bulb  from  the  bath,  dry  the 
exterior,  and  weigh.  Weigh  the  small  beaker  with  the 
liquid  contained.  Carefully  remove  the  liquid  from  the 
bulb  and  stem  as  described  above. 

The  difference  between  the  two  weights  of  the  bulb  when 
filled  with  liquid  gives  the  weight  M  —  m  of  liquid  expelled. 
The  difference  between  the  weight  of  the  flask  dry  and  after 
being  in  the  second  bath  gives  the  final  weight  of  liquid  in  the 
bulb.  The  expelled  liquid  is  saved  simply  as  a  check  and  is 
not  used  at  all  if  the  above  difference  be  slightly  greater.  1 

A  specific-gravity  bottle  may  be  substituted  for  the  bulb, 
but  is  not  as  satisfactory. 

Questions. 

1.  Why  is  double  weighing  unnecessary? 

2.  Why  is  M  —  m  determined  more  accurately  from  the  difference 
of  the  two  weighings  than  from  the  weight  of  the  liquid  expelled. 

3.  How  might  the  coefficient  of  expansion  of  a  solid,  attainable 
only  in  the  form  of  small  lumps,  be  found  by  an  extension  of  this 
method  ? 

4.  How  might  the  absolute  expansion  of  a  liquid  be  found  by  the 
above  apparatus? 

XVI.  COEFFICIENT    OF   INCREASE    OF   PRESSURE    OF 

AIR. 

Edser,  Heat,  pp.  106— in;  Poynting  and  Thomson,  Heat,  pp.  45-49; 
Text-book  of  Physics  (Duff),  pp.  188,  203-205;  Watson's  Pracr 
tical  Physics,  §§78,  79;  Watson's  Physics,  §§195—198;  Ames' 
General  Physics,  p.  240;  Crew's  General  Physics,  §269. 

If  the  volume  of  a  mass  of  gas  remains  constant  while 
its  temperature  is  raised,  its  pressure  increases  according 
to  the  law 


COEFFICIENT  OF  INCREASE  OF  PRESSURE  OF  AIR.    77 

in  which  P0  is  the  pressure  at  o°  C.,  P  the  pressure  at  the 
temperature  t,  and  a  is  a  constant  called  the  coefficient  of 
increase  of  pressure.  If  the  pressure  were  kept  constant 
and  the  volume  allowed  to  increase,  the  law  of  increase  of 
volume  would  be  similar,  and  it  is  found  that  the  constant 
a  is  practically  the  same  in  both  cases. 

It  is,  however,  difficult  to  keep  the  volume  exactly  con- 
stant, for  the  containing  vessel  will  expand  when  heated 
(the  volume  of  the  vessel  would  also  increase  because  of  the 
increase  of  pressure  to  which  it  is  subjected,  but  this  may 
be  neglected  since  it  is  extremely  small).  If  p  is  the  ob- 
served pressure  at  temperature  /  and  P0  the  observed  pres- 
sure at  o° 

/>  =  P0(i+a'*), 

where  a'  is  the  coefficient  of  apparent  increase  of  pressure 
(seeExp.  XV). 

To  correct  for  the  expansion  of  the  vessel,  we  must 
suppose  the  final  volume  of  the  gas  compressed  in  the  pro- 
portion in  which  the  capacity  of  the  vessel  expanded.  The 
law  of  expansion  of  the  vessel  is 

V  =  VQ     (l     +b     t), 

where  b  is  the  coefficient  of  cubical  expansion  of  the  vessel. 

To  get  the  pressure  P  that  would  keep  the  volume  of  the 

gas  absolutely  constant,  we  must  multiply  p  by   (i  +6  t), 

.'.  P=P0(i+a'/)(i+6/) 


And  so  the  true  coefficient  of  increase  of  pressure  a  is  ob- 
tained from  the  apparent  coefficient  of  increase  of  pres- 
sure a',  by  adding  the  coefficient  of  cubical  expansion  of  the 
vessel,  or, 

a  =  af  +  b. 

The  air  (or  gas)  is  enclosed  in  a  bulb  to  which  is  con- 
nected a  mercury  manometer.  The  pressure  indicated  by 
the  manometer  is  obtained  from  readings  of  the  mercury 
levels  on  a  scale  between  the  two  columns,  or,  preferably, 
with  a  cathetometer  (p.  19). 


HEAT. 


If  the  true  increase  of  pressure  of  dry  air  is  desired  the 
air  must  first  be  carefully  dried.  To  fill  the  bulb  with  dry 
air  it  may  be  connected  through  a  drying  tube  (containing 
chloride  of  calcium)  with  an  air-pump  and  the  bulb  several 
times  exhausted  and  refilled  with  air  sucked  through  the 
drying  tube.  (If  the  bulb  be  already  filled  with  dry  air 
the  process  will  be  unnecessary.) 

The  bulb  is  then  connected  to  the  manometer.  The 
bulb  is  first  immersed  in  a  bath  of  ice  and  water  as  nearly 
as  possible  at  o°,  and  the  movable  column 
of  the  manometer  is  adjusted  until  the 
mercury  in  the  other  column  is  at  a  definite 
point,  as  high  as  possible  without  entering 
the  contraction  where  connection  is  made 
with  the  bulb.  The  temperature  and  pres- 
sure are  read  as  carefully  as  possible,  at 
least  six  times,  when  both  have  become 
quite  steady,  the  manometer  being  read- 
justed before  each  reading. 

The  bath  of  ice  and  water  is  now  replaced 
by  one  of  water  at  about  10°  and  the  movable  column  is 
readjusted  until  the  mercury  in  the  stationary  column  is  at 
the  former  point,  that  the  volume  of  the  gas  may  remain 
constant.  The  temperature  and  pressure  are  read  when 
they  have  become  steady.  The  water  is  then  heated  to 
about  20°  and  the  observations  are  repeated.  Readings  are 
thus  made  at  intervals  of  about  i  o°  until  the  water  boils. 

The  pressure  and  temperature  when  the  water  is  boiling 
should  be  read  at  least  six  times,  the  mercury  level  in  the 
stationary  column  being  adjusted  to  the  constant  point  before 
each  reading.  It  is  at  the  initial  temperature  (near  o°)  and 
the  final  temperature  (near  100°)  that  the  most  reliable 
observations  are  obtained,  and  it  is  upon  these  that  the 
most  reliable  estimate  of  the  coefficient  of  expansion  is 
founded.  (The  readings  at  intermediate  temperatures  are 
made  in  order  to  test  the  law  of  expansion.)  If  the  two  arms 
of  the  manometer  are  of  different  radii,  there  will  be  a  con- 


FlG.     20. 


COEFFICIENT  OF  INCREASE  OF  PRESSURE  OF  AIR.    79 

stant  difference  of  level  due  to  capillarity.  This  should  be 
read  when  the  bulb  is  disconnected  and  allowance  should  be 
made  for  it  at  other  times.  Read  the  barometer  (p.  21)  and 
the  temperature  of  the  barometer  and  of  the  mercury  in  the 
manometer. 

Tabulate  from  your  observations  (a)  the  temperatures; 
(b)  the  differences  in  level  of  the  mercury  columns;  (c)  these 
differences  reduced  to  zero  degrees;  (d)  the  pressures  as 
calculated  from  (c)  and  the  barometer  heights  (reduced  to 
zero  degrees). 

The  test  of  the  law  of  expansion  is  made  by  plotting 
the  curve  of  pressure  and  temperature,  the  former  as  ordi- 
nates,  the  latter  as  abscissas.  This  should  be  nearly  a 
straight  line.  The  averages  for  the  first  point  (o°)  and  the 
last  (about  100°)  are  to  be  taken  as  fixing  the  straight  line. 
The  divergence  of  intermediate  points  from  the  straight  line, 
while  not  sufficient  to  invalidate  the  conclusion  that  the  in- 
crease of  pressure  is  linear,  will  illustrate  the  difficulty  of 
keeping  the  temperature  at  intermediate  points  constant 
for  a  sufficient  length  of  time  for  the  air  in  the  bulb  to  come 
wholly  to  the  temperature  of  the  water. 

Calculate  from  these  two  average  pressures  and  tem- 
peratures, the  coefficient  of  apparent  increase  of  pressure  (a') , 
and,  obtaining  the  coefficient  of  cubical  expansion  of  the 
glass  (6)  from  Table  VIII,  find  the  true  coefficient  of  increase 
of  pressure  (a) .  (Remember  that  the  coefficient  of  cubical 
expansion  is  three  times  the  coefficient  of  linear  expansion.) 

If  time  permit,  increase  the  range  of  temperature  by 
observations  below  o°  in  a  freezing  mixture  and  above  100° 
in  heated  oil. 

Questions. 

1 .  Why  must  (a)  the  air  be  dry  ?      (b)   a  capillary  connect  the  bulb 
and  the  manometer  ? 

2.  What  would  be  the  percentage  error  if  the  expansion  of  the 
bulb  was  neglected? 


8o 


HEAT. 


XVII.  PRESSURE     OF     SATURATED     WATER    VAPOR. 

Poynting  and  Thomson,  Heat,  Chap.  X;  Edser,  Heat,  pp.  220-228; 
Text-book  of  Physics  (Duff),  pp.  226-230;  Watson's  Physics, 
§§216-218;  Ames'  General  Physics,  pp.  264-269;  Crew' s  General 
Physics,  §§279-281. 

The  object  of  this  experiment  is  to  find  the  pressure  of 
saturated  water  vapor  at  different  temperatures.  By  pres- 
sure of  saturated  water  vapor  at  a  given  temperature,  or, 
as  it  is  often  called,  maximum  pressure  of  water  vapor,  or, 
equilibrium  pressure,  is  denoted  the  pressure  of 
the  vapor  above  water  in  a  closed  vessel  at  the 
given  temperature  after  a  steady  state  has  been 
reached.  A  liquid  continues  to  give  off  vapor 
from  the  surface,  or,  "evaporate,"  as  long  as  the 
pressure  of  the  vapor  above  the  liquid  is  less 
than  the  saturated  vapor  pressure,  independent 
of  the  total  atmospheric  pressure  above  the 
liquid.  After  the  pressure  of  the  vapor  reaches 
the  saturated  vapor  pressure  for  that  tempera- 
ture, the  total  quantity  of  vapor  in  the  atmos- 
phere above  the  liquid  remains  constant,  since 
for  any  vapor  given  off  from  the  surface  an  equal 
quantity  is  condensed. 

There  are  two  chief  methods  of  finding  the 
saturated  vapor  pressure,  the  static  method  and 
the  kinetic  method. 

(A)  In  the  static  method  some  water  (or  other  liquid)  is 
introduced  into  the  space  at  the  top  of  a  barometric  column 
which  is  surrounded  by  a  bath,  the  temperature  of  which 
can   be   varied.     The   pressure   of  the   vapor  is   found   by 
measuring  with   a  cathetometer  (p.   19)    the  height  of   the 
mercury  column  and  subtracting  this  from  the  barometric 
reading,  each  being  reduced   to   zero  (p.  21).      By  varying 
the  temperature  of  the  bath,  the  vapor  pressure  at  various 
temperatures  is  obtained. 

(B)  In  the  kinetic  method  the  quantity  measured  is  the 


FIG.  21. 


PRESSURE    OF    SATURATED    WATER    VAPOR. 


8l 


temperature  of  the  steam  above  water  boiling  under  different 
measured  pressures.  When  a  liquid  boils,  bubbles  of  vapor 
are  formed  throughout  the  interior  of  the  liquid.  In  forming 
these  bubbles,  the  vapor  overcomes  the  pressure  of  the 
atmosphere  above  the  liquid,  therefore  the  pressure  of  the 
vapor  must  equal  the  atmospheric  pressure,  and  obviously 
the  vapor  in  the  bubbles  is  saturated.  Hence,  in  measuring 
the  atmospheric  pressure  above  a  liquid  boiling  at  a  known 
temperature,  we  find  the  saturated  vapor  pressure  of  the 
liquid  at  this  temperature,  and  this  is  Regnault's  method, 
which  method  is  followed  in  this  experiment. 


FIG.  22. 

In  Regnault's  apparatus  the  total  pressure  above  the 
surface  of  the  liquid  can  be  kept  very  constant.  As  the 
liquid  is  heated,  the  vapor  is  condensed  in  a  Liebig  con- 
denser, and  as  the  pressure  of  vapor  distributed  through 
several  conducting  vessels  is  the  vapor  pressure  correspond- 
ing to  the  vessel  at  lowest  temperature,  the  pressure  exerted 
by  the  vapor  cannot  exceed  the  maximum  pressure  corre- 
sponding to  the  temperature  of  the  tap  water,  and  is  there- 
fore very  small.  As  the  temperature  of  the  boiler  changes, 
the  temperature  of  the  air  in  the  boiler  varies,  but  a  large 
air  reservoir  surrounded  by  water  is  connected  between  the 
condenser  and  the  manometer  and  air-pump  or  aspirator, 
which  makes  the  volume  of  the  air  in  the  boiler  small  com- 
6 


82  HEAT. 

pared  with  the  total  volume  of  air  in  the  system,  and  thus 
the  increase  of  pressure  due  to  the  heating  of  the  air  in  the 
boiler  is  small. 

The  boiler  should  be  about  two-thirds  full  of  water.  Fill 
with  water  the  small  tube  running  down  into  the  boiler 
(which  tube  is  closed  at  the  bottom),  and  insert  in  this 
tube  through  a  cork  one  of  the  thermometers  tested  by  the 
observer.  Draw  out  any  water  which  may  be  in  the  air 
reservoir  by  means  of  the  stopper  underneath.  Fill  the 
surrounding  vessel  with  water.  (Rubber  stoppers  should 
be  lubricated  with  rubber  grease  (note  p.  32)  before  inser- 
tion.) Exhaust  the  air  from  the  system  to  the  highest 
vacuum  attainable  by  means  of  a  Geryk  pump  or  aspirator. 
Close  all  the  cocks  through  which  connection  is  made 
to  the  aspirator  and  let  the  system  stand  a  few  minutes  to 
see  if  there  is  any  leakage.  If  not,  start  a  gentle  stream  of 
water  through  the  condenser,  and  place  a  Bunsen  flame 
under  the  boiler.  Read  the  barometer  and  its  temperature 
(see  p.  21). 

When  the  temperature  as  registered  by  the  thermometer 
in  the  boiler  becomes  very  steady,  read  it,  and  at  once 
record  the  two  extremities  of  the  mercury  column  of  the 
manometer.  Let  in  a  little  air  by  first  opening  and  then 
closing  a  cock  near  the  air-pump,  and  then  opening  and 
closing  a  cock  nearer  the  apparatus.  Increase  the  pressure 
at  first  by  about  15  mm.,  gradually  increasing  the  steps 
and  when  near  atmospheric  pressure  change  the  pressure 
by  about  1 2  cm.  The  reason  for  the  difference  in  pressure 
in  the  steps  is  that  it  is  better  to  have  the  steps  represent 
about  equal  changes  of  temperature,  for  instance,  about  5°. 

From  the  corrected  barometer  reading  and  the  differences- 
in  height  of  the  mercury  columns,  calculate  the  pressures. 
Tabulate   pressures  and  temperatures  and  also  plot  them, 
making  temperatures  abscissas  and  pressures  ordinates. 

Ramsay  and  Young's  method  for  measuring  the  vapor 
pressure  of  a  small  quantity  of  liquid  is  described  in  Wat- 
son's Practical  Physics,  §94. 


HYGROMETRY.  83 

Questions. 

1.  State  precisely  what  two  quantities  you  have  observed  in  the 
second  method  and  what  relation  they  bear  to  the  pressure  and  temp- 
erature of  saturated  vapor. 

2.  What  condition  determines  whether  a  liquid  will  boil  or  evapo- 
rate at  a  given  temperature? 

3.  What  was  the  actual  vapor  pressure  above  the  boiling  liquid? 
(Table  XIII.) 

4.  What  determines  (a)  the  lowest  temperature,  (b)  the  highest 
temperature  for  which  this  apparatus  is  applicable? 


XVIII.  HYGROMETRY. 

Poynting  and  Thomson,  Heat,  pp.  209-215;  Davis,  Elementary  Me- 
teorology, Chap.  VIII;  Robson,  Heat,  §§73—75;  Watson's  Prac- 
tical Physics,  §§95—97;  Text-book  of  Physics  (Duff),  pp.  239—242; 
Watson's  Physics,  §§220,  221;  Ames1  General  Physics,  pp.  265- 
268. 

Three  methods  will  be  used  for  studying  the  hygrometric 
state  of  the  atmosphere.  The  first  method  (A)  determines 
the  dew-point,  the  second  (B)  determines,  indirectly,  the 
actual  vapor  pressure,  and  the  third  (C)  determines  the 
relative  humidity. 

(A)  Regnanlt's  Hygrometer. — A  thin  silvered-glass  test- 
tube  is  half-filled  with  ether.  The  test-tube  is  tightly 
closed  by  a  cork  through  which  passes  a  sensitive  ther- 
mometer which  gives  the  temperature  of  the  ether.  Two 
glass  tubes  also  pass  through  the  cork,  one  extending  to  the 
bottom,  the  other  ending  below  the  cork.  An  aspirator 
gently  draws  air  from  the  shorter  tube.  The  ether  is 
evaporated  by  the  air  bubbles  and  the  entire  vessel  cools. 
The  silvered  surface  and  the  thermometer  are  watched 
through  a  telescope  and  the  temperature  is  read  the  moment 
moisture  appears  on  the  metal.  The  air  current  is  stopped 
and  the  temperature  of  disappearance  of  the  moisture  is 
observed.  This  is  repeated  several  times  and  the  mean 
is  taken  as  the  dew-point.  The  detection  of  moisture  is 
facilitated  by  observing  at  the  same  time  a  similar  piece 
of  silvered  glass  which  covers  a  part  of  the  test-tube,  but 
which  is  insulated  from  it.  The  temperature  of  the  air 


84  HEAT. 

should  also  be  carefully  determined,  preferably  with  a  ther- 
mometer in  a  similar  apparatus  where  there  is  no  evapora  - 
tion. 

An  arrangement  of  two  small  mirrors  at  right  angles 
so  placed  as  to  reflect  light  from  the  two  tubes  into  the 
telescope  will  facilitate  the  comparison. 

(B)  Wet  and  Dry  Bulb  Hygrometer. — Two  thermometers 
are  mounted  a  few  inches  apart.     About  the  bulb  of  one  is 
wrapped  muslin  cloth  to  which  is  attached  a  muslin  wick 
dipping  in  water.     The  other  is  bare.     The  temperatures 
of  both   are   read   when   they   have   become   steady.     The 
temperature  of  the  first  thermometer  will  be  lower  than  that 
of  the  bare  thermometer,  on  account  of  the  evaporation  of 
the  water.     From  the  difference  of  temperature  of  the  two 
thermometers  and  the  temperature  of  the  bare  thermometer 
the  actual  vapor  pressure  may  be  determined  with  the  aid 
of  empirical  tables    (see   Table   XV).     For  more   accurate 
apparatus,  see  references. 

(C)  Chemical   Hygrometer. — Fill    three    ordinary    balance 
drying   vessels   with   pumice.     Saturate    two    with    strong 
sulphuric  acid  and  the  third  with  distilled  water.     Weigh 
very  carefully  the  two  which  have  the  acid  and  then  connect 
them   to   an   aspirator,    with   the   water   absorption   vessel 
between  them.     After  a   gentle   stream  of  air  has  passed 
through  for  a  considerable  time,  disconnect  and  weigh  the 
sulphuric  acid  vessels.     The  ratio  of  the  gains  in  weight 
will  obviously  be  the  relative  humidity.     Observe  also  the 
temperature  of  the  air. 

If  not  directly  determined,  calculate  from  your  observa- 
tion, by  each  of  the  three  methods,  the  dew-point,  the 
actual  vapor  pressure,  the  relative  humidity,  and  the  amount 
of  moisture  in  the  atmosphere  per  cubic  meter.  Tabulate 
your  results.  Table  XIII  gives  the  vapor  pressures  of  water 
at  different  temperatures. 


SPECIFIC   HEAT    BY    THE    METHOD    OF    MIXTURE.  85 

XIX.  SPECIFIC  HEAT  BY  THE  METHOD  OF  MIXTURE. 

Edser,  Heat,  pp.  122-136;  Text-book  of  Physics  (Duff),  pp.  208—211; 
Watson's  Physics,  §§200-201;  Watson's  Practical  Physics,  §§82- 
84;  Ames'  General  Physics,  pp.  250-252;  Crew's  General  Physics, 
§252. 

The  specfic  heat  of  a  substance  is  the  number  of  calories 
required  to  raise  the  temperature  of  one  gram  of  the 
substance  one  degree  centigrade,  or  the  number  of  calories 
given  up  by  one  gram  in  cooling  one  degree  centigrade. 
In  the  method  of  mixture  a  known  mass  (M)  of  the  sub- 
stance, heated  to  a  known  temperature  (T),  is  immersed  in  a 
known  mass  of  liquid  (m)  of  known  specific  heat  (for  water 
=  i),  at  a  known  temperature  (/0),  and  the  unknown  mean 
specific  heat  (x)  of  the  substance  is  deduced  from  these  data 
and  the  temperature  (t)  to  which  the  mixture  rises.  Water 
is  the  liquid  employed  unless  there  would  be  a  chemical 
reaction  on  immersion. 

The  liquid  must  be  contained  in  a  vessel  which  is  also 
heated  by  the  immersion  of  the  hot  body.  The  heating  of 
the  vessel  is  equivalent  to  the  heating  of  a  certain  addi- 
tional quantity  of  water.  This  equivalent  quantity  of  water 
(e)  is  called  the  water  equivalent  of  the  vessel.  It  is  practi- 
cally equal  to  the  mass  of  the  vessel  (mj  multiplied  by  the 
specific  heat  (5)  of  the  material  of  the  vessel.  Theoretically 
it  may  be  obtained  by  noting  the  temperature  of  the  vessel 
and  pouring  into  it  a  known  mass  of  water  at  a  known 
temperature  and  noting  the  final  temperature.  This  is  an 
inverted  form  of  the  method  of  mixture  applied  to  finding 
the  specific  heat  of  the  vessel.  But  as  we  shall  presently 
see,  it  is  the  method  of  mixture  applied  under  very  unfavor- 
able conditions  and  will  not  usually  give  a  very  satisfactory 
result.  Another  method  will  be  recommended  below. 

The  equation  for  finding  the  specific  heat  is  obtained  by 
equating  the  heat  given  up  by  the  hot  body  to  that  taken  up 
by  the  water  and  containing  vessel.  Hence 

Mx(T-t)=(m+e)(t-t0).  (i) 


86  HEAT. 

Sources  of  Error. 

(1)  Loss  of  heat  while  the  hot  body  is  being  transferred 
to  the  water. 

(2)  Loss  of  heat  by  radiation,   conduction,   or  evapora- 
tion while  the  mixture  is  assuming  a  uniform  temperature. 

(3)  Errors  in  ascertaining  the  true  temperature  includ- 
ing errors  in  the  thermometers. 

Choice  of  Best  Conditions. — As  the  accuracy  of  this  de- 
termination depends  largely  on  the  selection  of  suitable 
conditions,  we  shall  consider  how  these  may  be  chosen  so 
that  unavoidable  errors  in  the  separate  measurements  may 
affect  the  result  as  little  as  possible. 

By  taking  the  logarithms  of  both  sides  of  (i)  and  dif- 
ferentiating partially,  we  obtain  (see  pp.  7,  8.) 

(2)  (3)  (4) 

f&cl          _8M      [~S*~|        Jm_      |~8*1         Be 

[He  \M~~W     [Hc\m~m+e      [x\e~m+e 

(5)  (6)  "  (7) 

M      *T   [H      *•    ry     (T-w 

[x\T  =      T-t'     UJ'o        *-*0'     Up     (T-t)(t-t0Y 

The  left-hand  side  of  (2)  stands  for  "the  ratio  that  the 
possible  error  (8#)  in  x,  due  to  the  possible  error  (8M)  in 
M,  bears  to  x,"  and  so  for  the  other  equations. 

M  and  m  can  be  measured  with  great  precision;  hence 
(2)  and  (3)  are  negligible.  From  (4)  it  is  seen  that  the 
water  equivalent  of  the  calorimeter  must  be  found  with 
some  care.  From  (5)  and  (6)  it  is  seen  that*  the  ranges 
T  —  t  and  t  —  tQ  should  be  as  great  as  possible  (see,  how- 
ever, "sources  of  error"  above).  This  is  also  consistent 
with  the  indications  of  (7),  for  although  (T  —  t0)  enters 
the  numerator,  the  product  of  T  —  t  and  t  —  tQ  is  in  the 
denominator.  Moreover,  it  is  seen  from  (5),  (6),  and  (7) 
that  if  equal  errors  are  made  in  observing  T,  t,  and  t0,  the 
effect  of  the  error  in  /  may  equal  the  sum  of  the  effects  of 
the  errors  in  T  and  t0.  Hence  the  necessity  of  determining 
t  with  special  care.  But,  allowing  an  unavoidable  error  in 


SPECIFIC   HEAT    BY    THE    METHOD    OF    MIXTURE.  87 

t,  how  can  its  effect  be  made  as  small  as  possible  by  prop- 
erly choosing  the  quantities,  M,  m,  t0,  Tf  Let  us  suppose 
T  —  /0  is  taken  as  great  as  possible  under  the  circumstances. 
How  can  (T  —  t)X  (*  — *0)  be  made  as  great  as  possible?  The 
sum  of  T  —  t  and  /  — 10  is  T  —  tQ,  a  fixed  quantity.  Hence  their 
product  is,  by  algebra,  a  maximum  when  they  are  equal  or 
t  is  midway  between  T  and  tQ.  But  it  is  seen  from  (i)  that 
this  also  requires  MX  and  m  +  e  to  be  equal.  Hence  we  see 
that  for  the  best  results,  T  and  t0  should  be  as  far  apart  as 
possible  and  the  heat  capacity  of  the  specimens  should  be  as 
nearly  as  possible  equal  to  the  heat  capacity  of  the  water  and 
the  vessel  that  contains  it. 

The  logical  procedure,  then,  would  be  to  roughly  de- 
termine x  by  the  method  of  mixture,  using  any  convenient 
values  of  M  and  m,  and  with  this  rough  value  for  x,  calcu- 
late what  ratio  of  M  to  m  would  best  satisfy  the  above  con- 
dition. Moreover,  it  is  seen  from  (4)  above  that  m  should 
be  as  large  as  is  consistent  with  other  conditions.  Then  we 
should  proceed  to  arrange  a  new  experiment  to  be  per- 
formed under  the  more  favorable  conditions  for  precision. 

We  now  see  why  it  is  not  easy  to  determine  the  water 
equivalent  of  the  vessel  directly.  Its  heat  capacity  is  small 
compared  with  that  of  the  water  that  would  fill  it,  and  so 
the  change  of  the  temperature  of  the  water  would  be  small 
and  difficult  to  determine  accurately.  If  a  much  smaller 
quantity  of  water  were  used,  a  large  part  of  the  surface  of 
the  vessel  would  be  left  uncovered,  and  its  temperature 
could  not  be  determined.  Hence  it  is  better  to  determine 
the  specific  heat  of  the  material  of  the  vessel,  using  the  ordi- 
nary method  of  mixture  and  a  mass  of  the  same  material  as 
the  vessel.  Then  multiplying  the  mass  of  the  vessel  by  its 
specific  heat,  we  have  its  water  equivalent. 

It  is  desirable  that  the  body  should  have  such  a  form 
that  it  and  the  water  in  which  it  is  immersed  should  rapidly 
come  to  a  common  temperature.  Filings,  shot,  thin  strips, 
wire  or  small  pieces  would  best  satisfy  this  condition. 
Larger  solid  masses  are  more  rapidly  (and  therefore  with 


88  HEAT. 

less  loss  of  temperature)  transferred  from  the  heater  to  the 
water,  and  to  give  the  water  ready  access  to  them  they  may 
be  perforated  with  holes,  through  which,  by  moving  the 
mass  up  and  down  in  the  water,  the  water  may  be  made  to 
circulate.  The  following  directions  apply  primarily  to  this 
latter  form,  but  may  be  readily  adapted  to  the  other  forms. 

Two  forms  of  heater  will  be  here  described,  (i)  The 
steam  heater.  A  copper  tube  large  enough  to  admit  the 
specimen  is  enclosed,  except  at  the  ends,  by  an  outer  copper 
vessel  which  is  to  act  as  a  steam  jacket  to  the  inner  vessel. 
Steam  from  a  simple  form  of  boiler  is  admitted  to  the 
jacket  through  a  tube  near  the  top  of  the  jacket  and  escapes 
through  an  outlet  near  the  bottom.  If  the  body  to  be 
heated  is  a  solid  mass,  it  is  suspended  in  the  heater  by  a  long 
string  that  passes  through  a  cork  that  closes  the  upper 
end  of  the  heater.  (If  the  specimen  is  in  the  form  of  shot 
or  clippings  they  are  placed  in  a  dipper  that  fits  into  the 
heater.)  A  thermometer  passed  through  the  cork  or  cover 
of  the  dipper  is  pushed  down  until  it  comes  into  contact  with 
the  body  tested.  The  lower  end  of  the  heater  is  also  closed 
with  a  cork. 

Such  a  steam  heater  ultimately  brings  the  specimen  to  a 
very  steady  temperature,  but  it  has  the  disadvantage  of 
heating  very  gradually.  If  the  boiler  which  supplies  steam 
to  the  jacket  has  a  closed  tube  extending  from  the  top  into 
the  interior  of  the  boiler,  of  slightly  larger  diameter  than 
the  specimen  or  dipper,  either  of  the  latter  may  be  placed 
therein  and  rapidly  heated  to  about  the  steam  temperature, 
when  they  are  transferred  to  the  steam  heater. 

(2)  The  Electric  Heater. — A  metallic  tube  is  heated  by 
a  strong  current  of  electricity  passing  through  a  coil  of  wire 
of  high  resistance  that  surrounds  the  tube.  The  current 
can  be  varied  by  changing  a  variable  resistance  in  circuit 
with  the  heating  coil.  With  an  alternating  current  the 
resistance  may  be  an  inductive  resistance  or  choking  coil 
consisting  of  wire  surrounding  a  soft-iron  wire  core.  A  low 
resistance  allowing  a  high  current  is  used  until  the  tern- 


SPECIFIC   HEAT    BY    THE    METHOD    OF    MIXTURE.  89 

perature  rises  to  the  desired  point  (perhaps  near  100°)  and 
then  the  current  is  reduced  to  the  strength  that  will  keep 
the  temperature  constant,  as  indicated  by  a  thermometer 
hung  in  the  heater.  The  body  is  introduced  into  the  heater 
exactly  as  in  the  case  of  the  steam  heater.  The  proper 
method  of  varying  the  resistance  can  only  be  learned  by 
some  practice. 

The  two  thermometers  used  should  be  those  for  which 
tables  of  corrections  have  been  obtained  earlier. 

The  calorimeter  may  be  prepared  while  the  specimen  is 
being  heated.  It  consists  of  a  smaller  copper  or  aluminum 
can  highly  polished  on  the  outside  and  enclosed  in  a  larger 
one  brightly  polished  on  the  inside,  but  well  insulated  from 
it  by  corks  or  cotton-wool.  A  wooden  cover  fits  over  both 
vessels  and  has  holes  for  thermometer  and  stirrer  and  an 
opening  giving  access  to  the  interior  vessel.  A  convenient 
form  has  a  trap-door  which  slides  open  in  two  halves, 
exposing  the  entire  inner  vessel.  A  screen  with  sliding  or, 
preferably,  double  swinging  doors,  should  separate  the 
calorimeter  from  the  boiler,  heater,  etc. 

The  "water  equivalent"  of  the  receiving  vessel  means 
the  water  equivalent  of  the  inner  vessel  together  with  the 
stirrer,  if  one  be  used.  It  is  advisable  that  the  stirrer 
should  also  be  of  copper  or  aluminum.  (A  stirrer  is,  how- 
ever, not  necessary  when  the  specimen  is  in  the  form  of  one 
large  block) . 

At  certain  times,  in  the  manipulation  of  this  experiment 
the  co-operation  of  two  persons  is  desirable,  and,  for 
economy  of  time,  two  determinations  should  be  made 
simultaneously,  two  heaters,  two  specimens,  and  two 
calorimeters  being  used.  One  specimen  should  peferably 
be  of  the  same  material  as  the  calorimeter,  so  that  the  water 
equivalent  may  be  determined. 

The  body  whose  specific  heat  is  to  be  determined  is 
weighed  to  o .  i  gm.  and  placed  in  the  heater  along  with  a 
thermometer.  The  inner  vessel  of  the  calorimeter  (including 
the  stirrer)  is  weighed  to  o.i  gm.  Water  near  the  tern- 


90  HEAT. 

perature  of  the  room  is  poured  into  it  until  it  is  judged  that 
when  the  hot  body  is  immersed  it  will  be  completely  covered 
and  the  water  will  rise  to  within  a  couple  of  centimeters  of 
the  top  of  the  vessel.  The  vessel  and  water  are  then 
weighed  to  o .  i  gm.  The  inner  vessel  is  now  replaced  in  the 
outer  and  the  cover  adjusted  and  closed. 

When  the  temperature  of  the  specimen  has  remained 
constant  for  ten  minutes,  it  may  be  assumed  that  the  hot 
body  is  throughout  at  the  temperature  of  the  heater.  The 
next  steps  require  two  persons,  and  as  it  is  important  that  it 
should  be  carried  out  promptly  and  neatly,  it  should  be 
carefully  considered  before  being  performed.  One  person 
constantly  stirs  the  water  in  the  calorimeter  and  reads  the 
temperature,  to  tenths  of  a  degree,  every  minute  for  five 
minutes.  He  then  opens  the  cover  and  slides  the  calorimeter 
beneath  the  heater.  The  other  observer  has  meanwhile 
made  a  careful  final  observation  of  the  temperature  of  the 
heater  and  removed  the  lower  stopper.  As  soon  as  the 
calorimeter  is  in  position,  he  lowers  the  hot  body,  without 
splash,  into  the  water.  The  calorimeter  must  then  be 
immediately  removed  and  the  cover  closed. 

One  observer  should  then  keep  the  mixture  stirred  by 
moving  the  body  up  and  down  with  the  aid  of  the  string 
and  note  the  temperature  at  as  short  equal  intervals  as 
possible  (perhaps  every  15  sees.)  while  the  other  records 
the  readings.  After  the  highest  temperature  has  been 
reached,  the  readings  are  continued  every  minute  for  at 
least  five  minutes. 

Simple  and  obvious  modifications  of  the  above  procedure 
are  required  if  the  specimen  is  in  the  form  of  shot  or  clippings. 

After  rough  calculations  of  water  equivalents  and  specific 
heats,  the  observers  should  exchange  duties  and  repeat 
the  whole,  using  masses  that  most  nearly  accord  with  the 
conditions  laid  down  in  the  considerations  stated  above. 
Before  calculating  final  results,  make  corrections  of  the 
temperatures,  T,  t,  t0,  according  to  the  tables  of  corrections 
already  obtained  for  the  thermometers  used. 


RATIO    OF    SPECIFIC   HEATS    OF    GASES.  91 

Plot  all  the  temperature  observations  of  the  final  deter- 
mination, and  correct  for  radiation  according  to  the  direc- 
tions given  on  pages  63,  64.  The  possible  error  of  the  result 
should  be  calculated  as  explained  on  pages  7,  8.) 

Questions. 

1.  How  might  the  present  method  be  adapted  to  find  the  specific 
heat  of  a  liquid? 

2.  Considering  evaporation,   loss   of   heat   when   transferring  the 
hot  body,  and  any  other  sources  of  error  that  may  occur  to  you,  is 
your  result  more  probably  too  high  or  too  low  ? 


XX.  RATIO  OF  SPECIFIC  HEATS  OF  GASES. 
(Clement  and  Desormes*  Method.) 

Edser,  Heat,  pp.  3 1 7-3 2 5 ;  Text-book  of  Physics  (Duff) ,  pp.  211-212, 
267,  268;  Watson's  Physics,  §§259-260;  Watson's  Practical 
Physics,  §105;  Ames'  General  Physics,  pp.  252-256. 

The  gas  is  compressed  into  a  vessel  until  the  pressure  has  a 
value  which  we  will  designate  by  pv  The  vessel  is  then 
opened  for  an  instant,  and  the  gas  rushes  out  until  the 
pressure  inside  falls  to  the  atmospheric  pressure,  pQ.  This 
expansion  may  be  made  so  sudden  that  it  is  practically 
adiabatic  and  the  temperature  of  the  gas  will  therefore 
fall.  After  the  vessel  has  been  closed  for  a  few  minutes,  the 
gas  will  have  warmed  to  the  room  temperature,  /,  and  the 
pressure,  p2,  will  be  above  that  of  the  atmosphere.  Consider 
one  gram  of  the  gas.  During  the  adiabatic  expansion,  its 
volume  changed  from  vl  to  v2,  according  to  the  adiabatic 
equation  for  pressure  and  volume  (see  references) 


W          Po 

Since  the  initial  and  final  temperatures  are  the  same,  and 
since  the  volume  remains  v2  while  the  gas  is  warming  and 
the  pressure  is  rising  from  p0  to  p2,  by  Boyle's  Law 


92  HEAT.. 

Hence  f,  the  ratio  of  specific  heats,  is  given  by  the  equation 


A  large  carboy  is  mounted  in  a  wooden  case  and  may  be 
surrounded  with  cotton  batting.  The  neck  is  closed  with 
a  rubber  stopper  through  which  passes  a  T-tube  connected 
on  one  side  with  a  compression  pump  (e.  g.,  a  bicycle  pump), 


FIG.  23. 

and  on  the  other  side  with^a  manometer  containing  castor 
oil.*  A  large  glass  tube,  which  may  be  closed  by  a  rubber 
stopper,  also  passes  through  this  large  stopper.  A  little 
sulphuric  acid  in  the  bottom  of  the  carboy  keeps  the  air 
dry.  A  very  fine  copper  wire  and  a  very  fine  constantin  wire 

*  The  density  of  castor  oil  is  about  .97,  but  it  should  properly  be  deter- 
mined (Exp.  VIII). 


RATIO    OF    SPECIFIC    HEATS    OF    GASES.  93 

pass  tightly  through  minute  holes  in  the  stopper  and  meet  at 
the  center  of  the  carboy,  in  a  minute  drop  of  solder. 

The  air  in  the  carboy  is  compressed  until  the  difference  in 
pressure  is  about  40  cm.  of  oil  (=Pi  —  po).  The  tube  connect- 
ing with  the  pump  is  closed,  and,  after  waiting  about  15 
minutes  to  allow  the  air  inside  to  regain  its  initial  tempera- 
ture (as  shown  by  the  pressure  becoming  constant),  the  ends 
of  the  oil  column  are  carefully  read.  The  carboy  is  now 
carefully  surrounded  with  cotton  batting,  which  may  have 
been  removed  to  facilitate  cooling.  The  air  inside  is  momen- 
tarily allowed  to  return  to  atmospheric  pressure  by  removing, 
for  about  one  second,  the  rubber  stopper  from  the  glass  tube. 
After  waiting  until  the  air  inside  has  assumed  the  room 
temperature  (shown  by  the  pressure  becoming  constant), 
the  final  pressure  p2  is  determined.  The  cotton-wool  had 
better  be  removed  during  this  stage. 

Connect  the  wires  to  a  calibrated  galvanometer  (Exp. 
LVIII),  apply  the  initial  compression  plt  and  observe  the 
reading  of  the  galvanometer  when  it  has  become  steady. 
Remove  the  stopper  as  before  (for  not  over  one  second) ,  re- 
place the  stopper,  and  observe  the  galvanometer  reading. 
The  proper  reading  to  record  is  the  fairly  steady  deflection 
which  is  attained  immediately  after  the  stopper  is  removed. 
There  are  liable  to  be  rapid  fluctuations  which  should  be 
disregarded,  and  of  course  the  temperature  does  not  long 
remain  steady,  owing  to  heating  or  cooling  from  the  outside. 
Record  as  before  the  final  pressure  p2.  Repeat  several  times, 
starting  with  the  same  initial  pressure  pv  Record  the 
temperature  of  the  room,  t,  and  p0,  the  height  of  the  baro- 
meter (p.  21). 

Calculate  y,  the  ratio  of  specific  heats  by  the  above  equation . 
Calculate  the  change  of  temperature  from  the  mean  of  the 
galvanometer  deflections  and  the  constants  of  the  thermo- 
couple and  galvanometer. 

Compare  the  result  with  Tl  —  T0  where  Tl  is  £  +  273  and 
ro  is  calculated  from  the  adiabatic  equation  for  temperature 
and  pressure  (see  references). 


94  HEAT. 


Unless  exceedingly  fine  wire  is  employed  (preferably 
No.  40,  B.  &  S.),  the  heat  capacity  of  the  wire  is  relatively 
so  great  that  the  thermocouple  will  not  show  the  full  change 
of  temperature. 

Draw  a  curve  with  volumes  as  abscissae,  and  pressures 
as  ordinates,  which  will  represent  the  changes  in  this 
experiment. 

(Let  specific  volumes,  i.  e.,  volumes  of  one  gram,  be  abscissae. 
Calculate  from  Table  VI  and  the  laws  of  gases  the  specific  volumes 
corresponding  to  the  room  temperature  and  p0,  plt  and  p2,  and  draw 
the  corresponding  isothermal.  Draw  the  horizontal  line  correspond- 
ing to  p0.  Draw  a  vertical  through  the  point  corresponding  to  p?  on 
the  above  isothermal.  The  intersection  of  these  two  straight  lines 
will  evidently  be  p0,  v2). 

Questions. 

1.  Do  you  see  any  objection  to  an  initial  exhaustion  of  the  gas 
in  place  of  the  compression? 

2 .  What  are  the  advantages  and  disadvantages  of  a  large  opening  ? 
Short  time  of  opening?     Castor  oil  manometer? 

3.  How  would  an  aneroid  manometer  be  preferable  in  this  experi- 
ment to  a  liquid  manometer? 


XXI.  LATENT  HEAT  OF  FUSION. 

Edser,  Heat,  pp.  145-149;  Text-book  of  Physics  (Duff),  p.  225; 
Watson's  Physics,  §211;  Watson's  Practical  Physics,  §88; 
Ames'  General  Physics,  pp.  260,  261;  Crew's  General  Physics, 
§286. 

The  latent  heat  of  fusion  of  a  substance  is  the  number  of 
calories  required  to  melt  one  gram  of  the  substance.  The 
most  common  method  of  measuring  it  is  a  method  of  mixture 
similar  to  that  used  in  finding  the  specific  heat  of  a  solid. 
A  known  mass  of  the  solid  at  its  melting-point  is  placed  in  a 
known  mass  of  the  liquid  at  a  known  temperature,  and  the 
temperature  of  the  liquid  observed  after  the  solid  has  com- 
pletely melted.  Allowance  must  be  made  for  the  water 
equivalent  of  the  calorimeter  and  correction  must  be  made 


LATENT  HEAT    OF    FUSION.  95 

for  the  effect  of  radiation  to  or  from  the  calorimeter  while 
melting  is  taking  place.  The  error  due  to  radiation  may  be 
made  small  by  having  the  liquid  initially  as  much  above 
the  temperature  of  its  surroundings  as  finally  it  falls  below. 
Thus  loss  and  gain  by  radiation  will  approximately  balance. 
Nevertheless,  since  the  calorimeter  will  probably  not  be  the 
same  length  of  time  above  the  temperature  of  the  sur- 
roundings as  below,  there  will  be  a  residual  error  for  which 
correction  must  be  made. 

The  calorimeter  consists  as  usual  of  an  inner  can  polished 
on  the  outside  to  diminish  radiation,  and  enclosed  in  an 
outer  can  polished  on  the  inside.  The  space  between  the 
two  cans  may  be  filled  by  cotton-wool  to  prevent  air  currents, 
and  still  further  prevent  communication  of  heat.  The  inner 
can  is  weighed,  first  empty  and  then  half-filled  with  warm 
water  about  1 5°  above  the  room  temperature.  It  is  then 
placed  in  the  outer  can  as  described  above  and  covered  by  a 
wooden  cover  having  holes  for  thermometer  and  stirrer 
and  a  hinged  cover  giving  access  to  the  inner  vessel. 

The  temperature  is  carefully  noted  each  minute  until  it 
has  fallen  to  about  10°  above  the  room  temperature.  In  the 
meantime,  ice  is  broken  to  pieces  of  about  a  cubic  centimeter 
in  volume.  These  pieces  are  carefully  dried  in  filter  paper. 
A  careful  observation  of  the  temperature  of  the  water  in 
the  calorimeter  having  been  made  and  the  time  noted,  a 
piece  of  ice  is  dropped  in  without  splashing  and  kept  under 
water  by  a  piece  of  wire  gauze  attached  to  the  stirrer. 
The  temperature  is  noted  every  half-minute  as  the  ice  melts, 
the  water  meantime  being  kept  stirred.  The  rate  at  which 
ice  is  dropped  in  is  regulated  simply  by  the  rate  at  which 
it  can  be  dried  and  the  temperature  and  time  noted. 

The  process  is  continued  until  the  temperature  has  fallen 
to  about  10°  below  the  room  temperature.  Then  the 
addition  of  ice  is  discontinued  and  the  temperature  of  the 
water  further  noted  every  minute  for  four  or  five  minutes. 
Finally,  the  weight  of  the  inner  can  and  its  contents  is 
obtained  in  order  that  the  mass  of  the  ice  may  be  deduced. 


96  HEAT. 

After  the  proper  weight  of  ice  has  been  ascertained,  the 
experiment  should  be  repeated  with  a  single  piece  of  approxi- 
mately this  weight.  As  there  are  considerable  sources  of 
error  that  cannot  be  eliminated,  the  whole  determination 
should  be  repeated  as  often  as  time  will  permit. 

All  the  temperature  observations  should  be  plotted 
against  the  time  and  the  radiation  correction  determined 
as  described  on  pages  63  and  64.  In  reporting,  consider 
the  possible  error  of  your  result  so  far  as  it  depends  on  the 
possible  error  of  your  weighings  and  observations  of  tem- 
perature. State  also  any  other  sources  of  error  that  may 
have  affected  your  result. 

Questions. 

1 .  What  advantages  are  there  in  the  use  of  one  large  lump  over  an 
equal  mass  of  small  ones? 

2.  Why  must  the  water  in  the  inner  vessel  be  pure? 

3.  Is  it  preferable  to  have  the  air  about  the  calorimeter  moist  or 
dry?     Explain. 


XXII.  LATENT  HEAT  OF  VAPORIZATION. 

Ames'  General  Physics,  p.  269;  Watson's  General  Physics,  §214;  Crew's 
Physics,  §287;  Watson,  Practical  Physics,  §§89-91;  Edser,  Heat, 
pp.  150-9;  Text-book  of  Physics  (Duff),  p.  231. 

The  latent  heat  of  vaporization  of  a  substance  is  the  number 
of  calories  required  to  change  one  gram  of  the  substance 
from  liquid  to  vapor.  The  usual  method  of  measuring 
it  is  a  method  of  mixture.  A  known  mass  of  vapor,  at  a 
known  temperature,  is  discharged  into  a  known  mass  of 
liquid,  at  a  known  initial  temperature,  and  the  final  tem- 
perature is  noted.  The  same  precautions  are  necessary 
as  in  finding  the  latent  heat  of  fusion.  The  arrangement 
of  the  calorimeter  is  also  the  same.  To  minimize  radiation 
the  water  should  be  initially  as  much  below  room  tempera- 
ture as  it  finally  rises  above,  say  15°.  The  initial  rate  of 
warming  should  also  be  obtained,  and  also  the  final  rate  of 
cooling. 


LATENT   HEAT    OF    VAPORIZATION. 


97 


Several  different  forms  of  boiler  have  been  devised  for 
the  purposes  of  this  determination.  Two  will  be  briefly 
described. 

In  Berthelot's  boiler  the  delivery  tube  passes  out  through 
the  bottom  of  the  boiler,  which  is  heated  by  a  ring  burner 
that  surrounds  the  tube.  Thus  the  tube  is  so  far  as  possible 
jacketed  by  the  boiling  water.  The  usual  form  of  this 

boiler  is  somewhat  fragile,  but  a 
good  substitute  may  be  made 
from  a  round-bottomed  boiling 
flask  the  neck  of  which  has  been 
shortened  (Fig.  24). 


FIG.  24. 


FIG.  25. 


In  the  electrically  heated  boiler  the  heating  of  the  water 
is  produced  by  a  coil  of  wire  that  is  immersed  in  the  water 
and  is  heated  by  a  current  of  electricity.  The  current  must 
be  kept  regulated  by  a  rheostat,  so  that  boiling  proceeds  at 
a  moderate  rate. 

The  chief  difficulty  is  in  delivering  the  steam  dry.  Con- 
densation is  apt  to  take  place  in  the  delivery  tube.  This 
can  be  reduced  by  inserting  a  trap  in  the  delivery  tube  be- 
tween the  boiler  and  the  calorimeter.  The  trap  should, 
from  time  to  time,  be  cautiously  heated  by  a  Bunsen  burner 
to  prevent  condensation,  but  in  general,  it  is  better  to  dis- 
pense with  the  trap  and  make  the  exposed  part  of  the 
7 


98  HEAT. 

delivery  tube  as  short  as  possible  and  carefully  cover  it  with 
cotton-wool. 

If  the  delivery  tube  simply  passed  to  a  sufficient  depth 
beneath  the  water,  the  steam  would  be  delivered  at  greater 
than  atmospheric  pressure,  as  the  pressure  of  a  certain  depth 
of  water  would  have  to  be  overcome.  Hence  it  is  better 
to  let  the  delivery  tube  pass  into  a  condensing-box  im- 
mersed in  the  water.  The  latter  must  also  be  open  to  the 
atmosphere  by  another  tube.  To  prevent  any  escape  of  steam 
by  this  tube  it  may  be  closed  by  a  little  cotton-wool.  The 
amount  of  steam  that  has  been  condensed  is  obtained  by 
weighing  the  condensing-box  (well  dried)  before  it  is  placed 
in  the  calorimeter,  and  again  with  the  contained  water 
at  the  end  of  the  experiment.  The  temperature  of  the  steam 
is  deduced  from  the  barometric  pressure.  A  pressure  gauge 
attached  to  the  boiler  affords  a  means  of  estimating  how  far 
the  pressure  differs  from  atmospheric  pressure. 

For  the  best  results,  certain  precautions  must  be  ob- 
served. The  delivery  tube  must  not  be  connected  to  the 
coridensing-box  until  steam  has  begun  to  pass  freely,  and 
as  dry  as  possible,  from  the  tube.  Connection  should  not 
be  attempted  until  the  temperature  of  the  water  has  been 
carefully  ascertained  and  care  has  been  taken  that  every- 
thing is  ready  for  making  a  deft  and  prompt  connection. 
After  the  temperature  of  the  well  stirred  water  in  the  inner 
calorimeter  has  been  read  every  minute  for  five  minutes,  the 
connection  is  made.  The  temperature  is  read  every  half- 
minute,  the  water  meantime  being  kept  well  stirred  by  a 
stirrer  (which  should  be  of  the  same  material  as  the  calor- 
imeter and  condenser  in  order  to  simplify  the  calculation  of 
the  water  equivalent).  The  flame  of  the  ring-burner  must 
be  regulated  so  that  the  steam  does  not  pass  too  rapidly.  This 
may  be  gauged  by  the  rate  of  the  rise  of  the  temperature 
of  the  calorimeter,  which  should  not  exceed  4°  or  5°  per 
minute.  In  finding  the  subsequent  rate  of  cooling,  the 
boiler  should  be  disconnected  from  the  condenser  and  the 
tube  leading  to  the  condenser  should  be  closed  by  plugs  of 


LATENT   HEAT    OF    VAPORIZATION.  99 

cotton-wool  to  prevent  evaporation;  but  in  subsequently 
weighing  the  condenser  the  wool  should  not  be  included. 
The  whole  determination  should  be  repeated  as  many  times 
as  possible. 

A  formula  for  the  calculation  of  the  latent  heat  may 
be  readily  worked  out.  Account  must  be  taken  of  the 
water  equivalent  of  calorimeter,  condenser,  and  stirrer.  The 
correction  for  radiation  is  made  by  the  method  stated 
on  pages  63,  64. 

The  possible  error  of  the  result,  so  far  as  it  depends  on 
the  readings  made,  should  be  calculated,  and  other  possible 
sources  of  error  should  be  mentioned. 

Questions. 

1 .  State  the  advantages  and  disadvantages  of  a  rapid  flow  of  steam  • 

2.  Explain  why  the  latent  heat  should  vary  with  the  atmospheric 
pressure. 

3.  Must  the  boiling  water  be  pure?     Explain. 


XXIII.  LATENT  HEAT  OF  VAPORIZATION. 

Continuous -flow  Method. 

Ames'  General  Physics,  p.  269;  Watson  s  Physics,  §214;  Crew1  s  General 
Physics,  §287;  Watson,  Practical  Physics,  §§89-91;  Edser,  Heat, 
pp.  150-9;  Text-book  of  Physics  (Duff},  p.  231. 

The  apparatus  for  this  method  may  be  readily  constructed 
from  a  Liebig's  condenser.  Water  enters  at  D  and  leaves  at 
C  through  T-tubes  connected  to  the  condenser  by  short 
rubber  tubes.  Superheated  steam  enters  at  A  through  a 
T-tube  and  the  condensed  water  drops  into  a  covered  beaker 
E.  The  steam  is  superheated  as  it  flows  through  a  glass 
tube  FE.  This  is  first  covered  with  asbestos  over  which  a 
heating  coil  of  wire*  is  wrapped,  the  coil  being  covered  by 
a  second  layer  of  asbestos.  AB  and  FE  are  mounted  on 
a  wooden  frame  and  A  B  is  thickly  covered  with  cotton- 
wool to  prevent  radiation.  Thermometers  7\,  T3,  T2,  T4, 

*  "Nichrome"  wire  (supplied  by  the  Driver-Harris  Co.,  New  York)  is  very 
suitable. 


IOO 


HEAT. 


give  the  respective  temperatures  of  the  superheated  steam, 
the  outflowing  water,  the  inflowing  water,  and  the  water  of 
condensation.  The  supply  of  water  may  come  from  the 
water  mains,  if  this  is  sufficiently  constant  in  temperature. 


FIG.  26. 

It  is,  however,  much  better  to  have  a  supply  of  from  5  to 
lo  gallons  in  an  elevated  tank  and  keep  the  flow  constant  by 
an  overflow  regulator  as  indicated  in  figure  29  (Exp. 
XXVIII). 

The  boiler  to  supply  the  steam  should  be  large  enough  to 
allow  of  a  flow  for  two  hours  without  refilling  (one  to  two 


LATENT   HEAT    OF    VAPORIZATION.  IOI 

liters  will  suffice).  The  current  in  the  superheating  coil 
should  be  regulated  by  a  rheostat  so  that  the  superheated 
steam  is  at  about  105°.  Some  time  should  be  spent  in  test- 
ing adjustments  to  obtain  a  suitable  current  and  a  rate  of 
flow  of  water  that  will  give  a  rise  of  temperature  of  about 
20°.  The  tank  should  be  connected  with  the  water  service 
so  that  it  can  be  readily  filled.  The  water  as  it  comes 
from  the  mains  will  probably  be  below  room  temperature 
and  this  is  an  advantage,  since  with  a  suitable  rate  of  flow 
of  the  steam  the  water  that  drops  into  E  will  differ  but  little 
from  room  temperature  and  will  suffer  little  loss  of  heat  by 
radiation.  This  will  require  a  proper  regulation  of  the 
burner  that  heats  the  boiler.  The  burner  should  be  sur- 
rounded by  a  shield  of  sheet-iron  or  asbestos  to  prevent 
fluctuations  caused  by  air-currents. 

The  thermometers  7\,  T2,  and  T3  should  be  read  once  a 
minute  (e.  g.,  T2  20  sec.  after  Tl  and  T3  20  sec.  after  T2) .  From 
the  mean  of  each  of  these  readings,  the  temperature  of  E, 
the  mass  of  water  that  flows  out  at  C,  and  the  mass  of  the 
water  that  drops  into  E,  the  latent  heat  can  be  calculated. 
The  specific  heat  of  the  superheated  steam  may  be  taken  as 
0.5.  A  formula  can  be  readily  constructed  to  express  the 
fact  that  the  heat  given  up  by  the  steam  and  condensed  water 
equals  the  heat  carried  off  by  the  current  of  water. 

Questions. 

1 .  Why  does  not  the  water  equivalent  of  the  condenser  need  to  be 
considered  ? 

2.  How  could  you  find  the  amount  of  error  due  to  conduction  of 
heat  from  the  superheater  to  the  water  in  the  condenser? 

3.  How  could  you  find  the  amount  of  error  due  to  radiation  from 
the  condenser? 

4.  What  other  sources  of  possible  error  are  there  in  this  method? 


IO2 


HEAT. 


XXIV.  THERMAL  CONDUCTIVITY. 

Edser,  Heat,  pp.  416-430;  Watson,  Practical  Physics,  §§106,  107; 
Text-book  of  Physics  (Duff),  pp.  216-220;  Watson's  Physics, 
§  §236—238;  Ames'  General  Physics,  p.  288;  Crew's  General  Physics, 
§§254,  256. 

The  thermal  conductivity  of  a  substance  is  the  amount 
of  heat  transmitted  per  second  per  unit  of  area  through  a 
plate  of  the  substance  of  unit  thickness,  the  temperature  of 
the  two  sides  differing  by  i°  and  the  flow  having  become 
steady.  If  K  be  the  thermal  conductivity,  and  if  a  plate 
of  thick-ness  /  and  area  A  be  kept  with  one  side  at  a  tem- 
perature t,  and  the  other  at  a  lower  temperature,  t',  the 
number  of  calories  that  will  flow  through  the  plate  in  time 
7,  after  the  flow  has  become  steady,  will  be 

KA(t-t')T 

I 

whence  K  can  be  derived  if  the  other  quantities  are  observed 
or  measured. 

Thermal  conductivity  is  in  general  difficult  to  measure 
satisfactorily.  The  following  very  simple  method  cannot 

be  relied  on  to  closer  than  a 
few  per  cent.,  but  it  only  re- 
quires a  small  portion  of  the 
time  that  the  more  accurate 
methods  call  for. 

A  rod  or  wire  of  the  substance 
to  be  tested  is  inserted  at  one 
end  into  a  heavy  block  of  metal, 
which  is  heated  to  a  constant 
high  temperature  in  a  bath, 
through  the  bottom  of  which 
the  rod  passes.  At  its  lower 
end  the  rod  is  screwed  into  a  heavy  block  of  brass  or 
copper  of  mass  M  and  specific  heat  s,  which  is  initially  at  a 
very  low  temperature.  Heat  is  thus  conducted  by  the  rod 
from  the  bath  to  the  lower  block.  If  the  latter  neither  lost 


FIG. 


THERMAL    CONDUCTIVITY.  103 

nor  gained  heat  by  convection  or  radiation,  and  if  there 
were  no  losses  from  the  sides  of  the  rod,  we  could  calculate 
the  conductivity  of  the  rod  from  its  dimensions  and  the 
mass,  specific  heat,  and  rise  of  temperature  of  the  lower 
block.  The  loss  of  heat  from  the  surface  of  the  rod  is 
almost  wholly  prevented  by  enclosing  it  by  a  glass  tube, 
which  does  not  come  into  direct  contact  with  the  rod,  and 
wrapping  the  glass  tube  with  cotton  wool  and  paper. 

To  allow  for  radiation  or  convection  to  or  from  the  lower 
block  the  experiment  is  modified  as  follows:  The  block 
is  enclosed  in  a  vessel  surrounded  by  a  water-jacket,  through 
which  water  at  a  constant  temperature,  t',  circulates.  Now, 
the  rate  at  which  the  lower  block  receives  heat  through  the 
rod,  when  the  former  is  at  the  temperature  tf,  is  the  mean 
of  the  rates  at  which  it  receives  heat  when  it  is  n  degrees 
below  t',  and  when  it  is  n  degrees  above  t' '.  For  let  R,  Rlt 
and  R2  represent  the  rates  of  conduction  of  heat  (flow  of 
heat  in  one  second)  to  the  lower  block  at  temperatures  t', 
t'  —  n,  and  tf  +n,  the  upper  end  being  at  temperature  /. 
Then 

KA(t-t') 


R 


I 

KA[t-(t'-n)] 
I 

KA[t-(t'+7i)] 


whence 


Again,  when  the  lower  block  is  at  the  same  temperature 
as  the  jacket,  it  neither  receives  heat  from  nor  gives  heat 
to  the  jacket.  And  when  it  is  n  degrees  below  it  gains 
heat  as  rapidly  as  it  loses  heat  when  it  is  n  degrees  above. 
Thus  by  taking  the  mean  rate  as  above,  the  effects  of  radia- 
tion to  or  from  the  block  are  eliminated.  In  fact,  adding 
a  to  Rlt  to  allow  for  the  gain  by  radiation,  and  subtracting 


104  HEAT. 

a  from  R2,  to  allow  for  loss  by  radiation,  would  leave 
unchanged. 

The  same  would  hold  true  for  any  other  pair  of  temper- 
atures equidistant  from  the  temperature  of  the  jacket. 
If  the  rates  of  rise  at  two  temperatures  equidistant  from 
the  temperature  of  the  jacket  be  rt  and  r2,  by  what  has  been 
said  the  rate  at  the  temperature  of  the  jacket  would  be 
i(ri+r2)-  Hence,  the  rate  at  which  the  body  must  be 
gaining  heat  is  Ms^(r1+r2).  Hence,  by  the  definition  of 
thermal  conductivity, 


or, 


A(t-t') 

The  lower  block  should  be  cooled  initially  to  about  12° 
below  the  temperature  of  the  water  that  circulates  through 
the  jacket  by  being  placed  in  a  bath  of  ice  and  water  (or 
snow).  When  taken  out,  it  must  be  carefully  dried.  The 
jacket  may  be  kept  at  a  constant  temperature  by  water 
passing  and  repassing  through  it  between  two  large  vessels, 
which  are  alternately  raised  and  lowered  about  every  five 
or  ten  minutes.  The  temperature  of  the  water  should  be 
frequently  read  by  a  thermometer  (which  may  conveniently 
pass  through  a  large  cork  that  floats  on  the  surface  of  the 
water).  If  the  temperature  of  the  water  should  show  a 
tendency  to  rise  or  fall,  a  small  quantity  of  cooler  or  warmer 
water,  respectively,  may  be  added.  If  the  vessels  be  large 
and  the  temperature  of  the  room  does  not  vary  widely, 
there  should  be  no  difficulty  in  keeping  the  water  constant  to 
within  .  2°  for  a  sufficient  length  of  time. 

The  hot  bath  is  in  the  form  of  a  trough,  which  is  heated 
at  one  end,  while  the  conducting  rod  passes  into  the  tank  at 
the  other  end.  To  prevent  direct  radiation  from  the  burner 
to  the  rod,  thick  screens  of  wood  and  asbestos  are  interposed. 
The  temperature  of  the  lower  block  should  be  read  at  least 


THE    MECHANICAL    EQUIVALENT    OF    HEAT.  105 

every  minute  by  means  of  a  thermometer  passing  through 
the  cork  or  fiber  cover  and  inserted  into  a  hole  in  the  block, 
the  unoccupied  space  in  the  hole  being  filled  with  mercury. 
The  readings  of  temperature  will,  from  various  causes,  be 
slightly  irregular.  They  should  therefore  be  plotted  in  a 
curve,  and  the  irregularities  eliminated  by  taking  the  more 
correct  values  from  the  curve.  The  temperature  of  the 
upper  block  should  be  read  frequently  by  a  thermometer 
thrust  into  it.  Small  quantities  of  boiling  water  should  be 
added  frequently  to  the  bath  to  compensate  for  evaporation. 

To  gain  some  idea  of  the  amount  of  reliance  to  be  placed 
on  the  result,  the  mean  rate  of  rise  for  each  pair  of  degrees 
equidistant  from  the  temperature  of  the  jacket  should  be 
obtained,  and  the  final  mean  of  all  taken  in  calculating. 
It  is,  however,  to  be  noted  that  the  temperatures  nearest 
the  jacket  temperature  should  give  the  best  results,  since 
there  the  radiation  is  a  minimum,  and  therefore  any  defect 
in  the  method  of  correcting  for  radiation  a  minimum. 

(For  comparing  the  conductivities  of  poorly  conducting 
substances  Lees  and  Chorlton's  apparatus  is  quite  satis- 
factory. Directions  for  its  construction  and  manipulation 
are  given  in  Robson's  Heat,  page  135.) 

XXV.  THE  MECHANICAL  EQUIVALENT  OF  HEAT. 

Griffith,  Thermal  Measurement  of  Energy,  Chap.  Ill;  Edser,  Heat, 
Chap.  XII;  Text-book  of  Physics  (Duff),  pp.  259-262;  Watson's 
Physics,  §§250-251;  Ames1  General  Physics,  pp.  203-205; 
Crew's  General  Physics,  §289;  Rowland,  Physical  Papers,  pp. 
343-476. 

The  mechanical  equivalent  of  heat  is  the  number  of  units 
of  mechanical  energy  that,  completely  turned  into  heat, 
will  produce  one  unit  of  heat,  or,  in  the  c.  g.  s.  system,  the 
number  of  ergs  in  a  calorie.  The  apparatus  here  described 
is  a  copy  of  that  used  in  the  University  of  Cambridge, 
England,  and  the  following  description  and  introduction 
is  partly  taken  from  that  issued  to  students  in  that  university. 


io6 


HEAT. 


In  this  apparatus  mechanical  energy  is  expended  in  working 
against  friction,  thus  producing  heat,  which  is  measured 
by  the  rise  in  temperature  of  a  known  mass  of  water. 

A  vertical  spindle  carries  at  its  upper  end  a  brass  cup. 
Into  an  ebonite  ring  concentric  with  the  cup  there  fits 
tightly  one  of  a  pair  of  hollow  truncated  cones.  The  second 
cone  fits  into  the  first,  and  is  provided  with  a  pair  of  steel 
pins  which  correspond  to  two  holes  in  a  grooved  wooden 


FIG.  28. 


disk,  which  prevents  the  inner  from  revolving  when  the 
spindle  and  the  outer  cone  revolve.  A  cast-iron  ring, 
resting  on  the  disk  and  fixed  by  two  pins,  serves  to  give  a 
suitable  pressure  between  the  cones.  A  brass  wheel  is 
fixed  to  the  spindle,  and,  by  a  string  passing  round  this 
wheel  and  also  round  a  hand-wheel,  motion  is  imparted  to 
the  spindle.  A  pair  of  guide  pulleys  prevents  the  string 
from  running  off  the  wheel.  Above  the  wheel  is  a  screw 
cut  upon  the  spindle.  This  screw  actuates  a  cog-wheel  of 
100  teeth,  which  makes  one  revolution  for  every  100  revo- 
lutions of  the  spindle. 


THE    MECHANICAL    EQUIVALENT    OF    HEAT.  107 

To  the  base  of  the  apparatus  one  end  of  a  bent  steel  rod 
is  attached;  the  rod  can  be  fixed  in  any  position  by  a  nut 
beneath  the  base.  The  other  end  of  the  rod  carries  a  cradle, 
in  which  runs  a  small  guide  pulley  on  the  same  level  with  the 
disk.  The  cradle  turns  freely  about  a  vertical  axis.  A  fine 
string  is  fastened  to  the  disk  and  passes  along  the  groove  in 
its  edge;  it  then  passes  over  the  pulley  and  is  fastened  to  a 
mass  of  200  or  300  grams.  On  turning  the  hand- wheel  it  is 
easy  to  regulate  the  speed  so  that  the  friction  between  the 
cones  just  causes  the  mass  to  be  supported  at  a  nearly 
constant  level.  To  prevent  the  string  from  running  off 
the  guide  pulley,  a  stiff  wire  with  an  eye  is  fixed  to  the 
cradle  and  the  string  is  passed  through  this  eye.  It  also 
passes  through  an  eye  fixed  to  the  steel  rod,  to  prevent 
the  weight  from  being  wound  up  over  the  pulley. 

The  rubbing  surfaces  of  the  cones  must  be  carefully 
cleaned  and  then  four  or  five  drops  of  oil  must  be  put  be- 
tween them;  the  bearings  of  the  spindle  and  guide  pulley 
should  also  be  oiled.  The  cones  are  then  weighed  together 
with  the  stirrer.  The  inner  cone  is  then  filled  to  about  i 
cm.  from  its  edge  with  water  2°  or  3°  below  the  temperature 
of  the  room  and  the  system  is  again  weighed.  The  cones 
are  then  placed  in  position  in  the  machine  and  a  thermometer 
is  hung  from  a  support  so  that  it  passes  through  the  central 
aperture  in  the  disk  and  almost  touches  the  bottom  of  the 
inner  cone. 

One  observer,  X,  takes  his  place  at  the  hand-wheel,  and 
the  other,  Y,  at  the  friction  machine.  By  working  the 
machine  the  water  is  now  warmed  up  until  its  temperature  is 
nearly  equal  to  that  of  the  room.  The  index  of  the  counting 
wheel  is  read  and  the  temperature  of  the  water  is  carefully 
observed  every  minute  for  five  minutes.  Immediately  after 
the  last  reading,  X  turns  the  wheel  fast  enough  to  raise  the 
mass  until  the  string  is  tangential  to  the  edge  of  the  disk. 
If  the  string  be  not  tangential  the  moment  of  its  tension 
about  the  axis  of  revolution  is  seriously  diminished.  Y 
stirs  the  water  and  notes  the  temperature  at  each  passage 


108  HEAT. 

of  the  zero  of  the  counting-wheel  past  the  index;  each 
passage  of  the  zero  after  the  first  corresponds  to  100  revolu- 
tions of  the  spindle.  Y  gives  a  signal  at  each  passage  of  the 
zero  and  X  notes  the  time  by  aid  of  a  watch.  After  Y  has 
recorded  the  temperature  upon  a  sheet  of  paper  previously 
ruled  for  the  purpose,  he  also  records  the  time  observed  by 
X.  After  about  1000  revolutions  the  motion  is  stopped 
and  the  readings  of  the  index  of  the  counting  wheel  and  of  the 
thermometer  are  recorded.  Observations  of  the  temper- 
ature are  continued  every  minute  for  five  minutes,  the  stirring 
of  the  water  being  continued. 

The  temperature  observations  are  plotted  against  the 
time,  and  the  radiation  correction  is  determined  as  explained 
on  pages  63  and  64.  The  heat  produced  is  readily  cal- 
culated from  the  mass  of  water,  the  water  equivalent  of 
the  cones  and  stirrer,  and  the  corrected  rise  of  temperature. 

From  the  initial  and  final  readings  of  the  counting  wheel 
and  the  number  of  complete  revolutions  the  exact  number 
(n)  of  revolutions  made  by  the  spindle  is  deduced.  The 
work  done  is  calculated  as  follows:  When  the  spindle 
has  made  n  turns  the  work  spent  in  overcoming  the  friction 
between  the  cones  is  the  same  as  would  have  been  spent 
if  the  outer  cone  had  been  fixed  and  the  inner  one  had  been 
made  to  revolve  by  the  descent  of  the  mass  of  M  grams. 
In  the  latter  case  M  would  have  fallen  through  innr  cm. 
where  r  is  the  radius  of  the  groove  of  the  wooden  disk,  which 
must  be  measured.  Hence  the  total  work  spent  against 
friction  and  turned  into  heat  is  2nnrMg  ergs.  In  the  report, 
estimate  the  possible  error  of  the  result  as  far  as  it  depends 
upon  the  errors  of  observations  and  measurements. 


Questions. 

1 .  What  amount  of  error  is  due  to  neglect  of  the  work  spent  against 
friction  of  the  bearing  of  the  outer  cone  ? 

2.  Why  must  the  wheel  be  turned  faster  as  the  experiment  pro- 
ceeds? 

3.  What  effect  on  the  result  has  the  variation  of  the  viscosity  of 
the  oil? 


THE    MELTING-POINT    OF   AN   ALLOY.  IOQ 

XXVI.  THE  MELTING-POINT  OF  AN  ALLOY. 

Robson,  Heat,  pp.  77-79;  Findlay,  Phase  Rule,  pp.  220-223;  Ewell, 
Physical  Chemistry,  pp.  271-272. 

If  an  alloy  is  melted  and  is  allowed  to  cool  while  its  tem- 
perature is  continuously  observed,  and  a  curve  be  then  drawn 
with  times  as  abscissae  and  temperature  as  ordinates,  it 
will  be  found  that  at  certain  points  the  curvature  abruptly 
changes,  the  fall  of  temperature  being  decreased  or  even 
ceasing.  At  the  moment  corresponding  to  such  a  point,  the 
alloy  is  radiating  heat  to  the  room,  and  the  fact  that  its 
temperature  does  not  fall  as  rapidly  indicates  that  heat  is 
being  produced  internally  by  some  change  of  state  of  the 
material.  Such  a  point  is  therefore  a  solidifying-point  of 
some  constituent  of  the  alloy  or  of  the  eutectic  alloy. 

The  assigned*  metals  are  carefully  weighed  and  melted  in 
an  iron  cup.  A  copper-constantinf  thermocouple  is  plunged 
into  the  liquid  metal  and  kept  there  until  the  entire  mass  is 
solid.  A  porcelain  tube  should  cover  one  wire  for  some 
distance  from  the  junction.  The  terminals  are  connected 
to  a  calibrated  galvanometer  through  a  resistance  such  that 
the  maximum  deflection  will  keep  on  the  scale.  The  galvan- 
ometer is  read  every  half-minute  and  the  time  of  each  read- 
ing is  noted.  When  the  readings  are  commenced,  the  metal 
should  be  considerably  above  the  melting-point  and  the 
readings  should  be  continued  for  some  time  after  the  metal 
is  apparently  solid.  For  calibration  of  the  galvanometer,  see 
Exp.  LVIII. 

Plot  the  galvanometer  deflections  against  the  time.  De- 
termine the  electromotive  force  corresponding  to  the  galvan- 
ometer deflections  where  the  curvature  changed,  and  from 
the  constants  of  the  thermocouple,  or  a  chart  giving  the 

*  Tin  and  lead  are  suitable  metals.  The  changes  of  curvature  are  more 
distinct  if  the  former  is  in  excess.  The  eutectic  of  tin  and  lead  is  composed 
°f  37%  lead,  63%  tin,  and  melts  at  182.5°  (Rosenhain  and  Tucker,  Roy.  Soc. 
Phil.  Trans.,  1908,  A.  209,  p.  89). 

f  "Advance"  Wire,  Driver-Harris  Co.,  Harrison,  N.  J. 


1 10  HEAT. 

temperature   for   different   electromotive   forces,    determine 
the  temperature  of  these  points. 

Tabulate  the  observed  temperatures  of  these  transition 
points  and  your  opinion  of  what  they  represent. 

Questions. 

1.  Explain   why  the   second   transition   point   is   represented   by 
a   horizontal   portion  of  the  cooling  curve  while  the  first  transition 
point  is  merely  represented  by  a  change  of  curvature. 

2.  Will  the  temperature  of  the  first  point  vary  with  the  initial 
concentration  ? 

3.  Will  the  temperature  of  the  second  transition  point  vary  with 
the  initial  concentration  ? 


XXVII.  HEAT  VALUE  OF  A  SOLID. 

Ferry  and  Jones,  pp.  237-242. 

(A)   HEMPEL  BOMB  CALORIMETER  (Constant-volume 
Calorimeter) 

A  pellet  of  the  fuel  to  be  tested  is  formed  in  a  press,  a 
cotton  cord  being  imbedded  with  a  loose  end.  After  being 
pared  down  to  about  i  gm.  and  brushed,  it  is  carefully 
weighed.  It  is  then  suspended  in  a  Hempel  combustion 
bomb,  and  the  thread  is  wrapped  around  a  platinum  wire 
connecting  the  platinum  supports  of  the  basket.  The  ter- 
minals attached  to  these  supports  are  connected  with  several 
Edison  or  storage  cells  sufficient  to  just  bring  the  wire 
to  a  brilliant  incandescence  (as  ascertained  by  a  preliminary 
trial) . 

The  bomb  is  charged  with  oxygen  under  at  least  fifteen 
atmospheres'  pressure,  either  from  a  charged  cylinder  or 
produced  by  a  retort.  Bomb  and  pressure  gauge  should  be 
immersed  in  water  while  the  oxygen  is  being  supplied.  As- 
certain that  the  bomb  valve  is  open  and  that  all  connections 
are  screwed  tight.  Open  the  cylinder  valve  (if  a  cylinder  is 
used)  until  the  pressure  becomes  high,  and  then  close.  Lift 
the  bomb  out  of  the  water,  loosen  one  of  the  connections, 
and  allow  the  mixture  of  air  and  oxygen  to  escape;  then 


HEAT  VALUE  OF  A  SOLID.  Ill 

tighten,  replace  in  water,  open  the  cylinder  valve  again  until 
the  pressure  becomes  high  (at  least  fifteen  atmospheres); 
close  both  the  cylinder  valve  and  that  of  the  bomb,  and 
finally  disconnect  and  dry  the  bomb.  (If  the  oxygen  is 
produced  in  a  retort,  partly  fill  the  latter  with  a  five  to 
one  mixture  of  potassium  chlorate  and  manganese  dioxide, 
connect  to  the  bomb  and  pressure  gauge,  and  heat  the  upper 
part  slowly  with  a  Bunsen  burner.) 

Attach  the  electrical  terminals,  place  the  bomb  in  the 
special  vessel  containing  about  a  liter  of  water,  adjust  the 
Beckmann  thermometer  to  read  about  i°  (see  p.  65),  stir 
the  water  continually,  and  read  its  temperature  every  half- 
minute  for  five  minutes,  estimating  to  tenths  of  the  smallest 
graduation.  Close  the  electric  switch;  after  a  few  seconds 
open  it  and  read  the  temperature  of  the  continually  stirred 
water  for  ten  minutes. 

Let  H  be  the  heat  value  of  the  fuel  and  m  the  mass  of 
the  specimen,  M  the  mass  of  the  water,  e  the  water  equivalent 
of  the  bomb,  tl  the  initial  temperature,  and  t2  the  final  tem- 
perature (corrected  for  radiation,  see  p.  63.)  Then 

eWi-tJ 

—  cal.  pergm. 
wi 

The  best  method  in  practice  to  determine  e  is  to  repeat 
the  determination,  using  'salicylic  acid  as  fuel,  and  assum- 
ing its  heat  value  to  be  5300  calories  per  gram. 

(B)  ROSENHAIN'S  CALORIMETER  (Constant-pressure 
Calorimeter) 

Phil.   Mag.,   VI,  4,  p.  451. 

Instead  of  burning  the  fuel  in  a  fixed  volume  of  highly 
compressed  oxygen,  the  oxygen  is  supplied  continuously 
at  only  slightly  above  atmospheric  pressure. 

The  coal  is  pulverized  and  a  sample  is  compressed,  in  a 
special  screw  press,  into  a  pellet  weighing  about  one  gram. 
This  is  placed  on  a  porcelain  dish  which  rests  on  the  bottom 


112 


HEAT. 


of  the  inside  chamber.  The  ignition  wire  should  be  about 
3  cm.  of  No.  30  platinum  wire  and  the  external  terminals 
should  be  connected  to  storage-battery  terminals  through  a 
key  and  a  resistance  such  that  the  wire  will  glow^brightly. 
A  gasometer  is  charged  with  oxygen  from  a  cylinder  or  gen- 
erated from  "oxone"  and  water.  The  action  of  the  dif- 
ferent valves  having  been  studied,  the  apparatus  should 
be  assembled,  the  upper  side  valve  (see  Fig.  29)  being  closed 
and  the  ball  valve  lowered.  Connect  with 
the  oxygen  supply  through  a  wash-bottle, 
turn  on  a  very  gentle  stream  of  oxygen, 
and  pour  into  the  outer  vessel  a  measured 
volume  of  water,  at  the  room  tempera- 
ture, so  that  the  combustion  chamber  is 
just  covered. 

If  a  Beckmann  thermometer  (see  p.  65) 
is  used,  adjust  to  read  between  o°  and  i° 
in  this  water.  The  bulb  should  be  sup- 
"~\CT_-Jj/  ported  on  a  level  with  the  center  of  the 
combustion  chamber.  Read  the  tempera- 
ture every  half-minute  for  five  minutes, 
then  increase  the  oxygen  current,  and  care- 
fully read  the  temperature  and  the  time,  close  the  key  and 
ignite  the  pellet  with  the  hot  platinum  wire,  and  immediately 
remove  the  wire.  During  such  operations  it  is  best  to  hold 
the  inner  vessel  steady  by  grasping  the  oxygen  inlet  tube. 
Keep  the  water  pressure  in  the  gasometer  constant,  and  as 
combustion  proceeds  increase  the  flow  of  oxygen.  If  pos- 
sible, read  the  thermometer  every  half-minute. 

When  combustion  has  ceased,  move  the  hot  wire  about 
to  igni-te  any  unconsumed  particles.  Keep  the  wire  hot  as 
short  a  time  as  possible  and  remove  it  immediately  from  any 
combustion,  otherwise  it  is  liable  to  be  melted. 

Finally,  turn  off  the  oxygen  supply,  open  the  upper  valve, 
and  raise  the  ball  valve,  allowing  the  water  to  enter  the 
inner  chamber.  Then  force  out  the  water  by  closing  both 
valves  and  turning  on  the  oxygen.  Record  the  highest 


FIG.  29. 


HEAT    VALUE    OF   A    GAS    OR    LIQUID.  113 

temperature  and  the  time  and  the  temperature  every 
half-minute  for  five  minutes.  For  radiation  correction, 
see  p.  63,  and  for  formulae,  see  (A)  preceding. 

To  determine  e,  assemble  the  apparatus,  including  the 
Beckmann  thermometer,  and  pour  in  1000  c.c.  of  water. 
Determine  very  carefully  the  temperature  with  a  0.1° 
thermometer  and  then  add  500  c.c.  of  water  at  about  50°,  the 
temperature  of  which  has  also  been  very  carefully  deter- 
mined. Determine  also  very  carefully  the  final  steady 
temperature  and  from  these  data  determine  e. 

For  anthracite  coal  add  sugar  in  the  proportion  3:1. 
(Heat  of  combustion  of  sugar  =  3 900  calories  per  gram.) 

Questions. 

1.  Calculate  the  heat  value  of  (a)  one  kilo  of  this  substance,  (b) 
one  short  ton  in  B.  T.  U.  per  lb.,  (c)  the  mechanical  energy  equivalent 
to  the  latter. 

2.  What  error  would  be  caused  by  (a)  an  error  of  20  in  the  water 
equivalent?      (b)  allowing  a  current  of  5  amperes  to  flow  through  a 
platinum  wire  of  2  ohms'  resistance  for  5  seconds?      (c)  Neglecting 
the  radiation  correction  ? 

XXVIII.  HEAT  VALUE  OF  A  GAS  OR  LIQUID. 

Ferry  and  Jones,  pp.  243-246. 

The  heat  value  will  be  determined  with  Junker's  cal- 
orimeter. 

(A)  A  measured  volume  (v  liters)  of  gas  under  an  ob- 
served pressure,  p,  is  burned  in  the  calorimeter,  and  the 
rise  of  temperature,  from  tj°  to  t2°,  of  a  mass  of  M  gr.  of 
water,  is  determined.  The  flow  of  water  and  gas  is  so  regu- 
lated that  the  burned  gas  leaves  the  calorimeter  at  approxi- 
mately the  temperature  of  the  entering  gas,  and  there 
should  be  a  difference  of  at  least  6°  in  the  temperature  of  the 
in-  and  outflowing  water.  Also,  the  flow  of  water  must  be 
sufficient  to  furnish  a  constant  small  overflow  at  the  supply 
reservoir  (see  Fig.  30).  The  burner  should  be  lighted  out- 
side the  calorimeter.  When  the  temperatures  indicatedjDn 
the  various  thermometers  have  become  constant,  note  the 
gasometer  reading,  and  immediately  collect  in  graduates  the 
8 


HEAT. 


heated  overflowing  water,  and  also  the  water  condensed 
by  the  combustion  of  the  gas.  Let  the  mass  of  the  latter  be 
m  gr.,  and  its  temperature  /'.  Note  the  temperatures  of  the 
inflowing  and  outflowing  water  every  15  sec.  until  two  or 

three  liters  have  passed  through. 
Then  immediately  note  the  gaso- 
meter reading  and  remove  the 
graduates. 

Assuming  that  condensation  of 
the  gas  occurs  at  100°,  the  heat 
liberated  is  m  [536+100  —  2'].  If 
H  represents  the  heat  value  of 
the  gas  in  gram-calories  per  liter 
and  v  the  volume,  reduced  to  o°, 
and  760  mm., 

M(t2-t1)-m(636-tf) 


FIG.  30.  (B)   To     determine     the     heat 

value    of    a   liquid   fuel,   the    gas 

burner  is  replaced  by  a  suitable  lamp  which  is  attached  to 
one  arm  of  a  balance.  The  rate  at  which  the  liquid  is 
consumed  is  determined  from  the  weights  in  the  pan,  on  the 
other  side,  at  different  times.  It  is  best  to  make  the  weight 
in  the  pan  slightly  deficient  and  note  the  exact  time  when 
the  balance  pointer  passes  zero,  as  the  liquid  is  consumed. 
Practically  complete  combustion  is  obtained  with  a  "Primus" 
burner,  supplied  from  a  reservoir  where  the  liquid  is  under 
considerable  pressure.  With  very  volatile  liquids,  the 
opening  of  the  burner  must  be  large  and  the  pre-heating 
tubes  must  be  in  the  cooler  part  of  the  flame. 

Express  your  results  in  (a)  calories  per  liter  (b)  B.  T.  U. 
per  gallon  or  cubic  foot. 

Questions.  « 

1.  Is  the  heat  value  of  a  gas,  in  calories  per  gram,  definite?    Per  liter? 

2.  What  difference  would  there  be  in  the  result  if  all  the  water 
vapor  escaped  without  condensing? 

3.  Why  is  no  radiation  correction  necessary? 


PYROMETRY.  115 

XXIX.  PYROMETRY. 

Edser,  Heat,  pp.  339—411;  Bulletin,  Bureau  of  Standards,  I,  2,  pp. 
189-255;  Watson,  Practical  Physics,  §§208-210;  Le  Chatelier, 
High  Temperature  Measurements,  Chaps.  Ill,  VI,  IX. 

This  exercise  is  a  study  of  three  of  the  methods  used  in 
the  measurement  of  very  high  temperatures. 

A  hollow  black  body  is  to  be  heated  electrically  and  the 
temperature  of  its  interior  is  to  be  determined  by  means 
of  a  calibrated  thermocouple.  A  platinum  resistance  ther- 
mometer is  to  be  calibrated,  and  also  an  incandescent  lamp 
is  to  be  calibrated  for  use  as  an  optical  pyrometer. 

Electric  Furnace. — The  electric  furnace  (see  Fig.  31) 
consists  of  a  thin  porcelain  cylinder  about  15  cm.  long  and 


FIG.  31. 

10  cm.  in  diameter  upon  which  is  wound  about  5  m.  of 
No.  22  "  Nichrome"  wire,*  if  a  220-volt  supply  is  to  be  used. 
Whatever  the  voltage,  the  winding  must  be  such  as  to 
consume  about  half  a  kilowatt.  The  ends  of  the  cylinder 
are  closed  by  porcelain  caps  with  proper  apertures  and  the 
whole  is  surrounded  by  many  layers  of  asbestos. 

Heating  and  cooling  must  be  very  gradual  so  that  the 
thermocouple     and     platinum    thermometer    may    acquire 
the  temperature  of  the  furnace.     The  highest  temperature 
should  not  exceed  1000°. 
*$ee  note  bottom  of  page  99, 


Il6  HEAT. 

Thermocouple. — A  platinum  and  platinum  +10%  rhodium 
thermocouple  should  be  connected  to  a  galvanometer  through 
a  key  and  such  a  resistance  as  will  keep  the  deflection  on  the 
scale  at  the  highest  temperature.  The  galvanometer  with 
resistance  should  be  calibrated  as  a  voltmeter  (Exp.  LVIII). 
The  chart  or  table  accompanying  the  couple  gives  the 
temperature  of  the  hot  junction  when  the  electromotive 
force  is  known.  (The  cool  junction  should  be  in  ice  and 
water.) 

Platinum  Resistance  Thermometer. — The  platinum  resist- 
ance thermometer  consists  of  a  coil  of  fine  platinum  wire 
(for  example,  50  cm.  of  No.  30)  wound  on  a  porcelain  frame 
and  surrounded  by  a  glazed  porcelain  tube.  The  coil 
constitutes  one  arm  of  a  Wheatstone's  bridge.  A  pair  of 
dummy  leads  are  connected  to  an  adjoining  arm  (see  figure) 
and  compensate  for  the  heating  of  the  lead  wires.  A 
suitable  switch  connects  the  galvanometer  to  either  the 
bridge  or  the  thermocouple. 

Optical  Pyrometer. —  The  optical  pyrometer  consists 
essentially  of  a  lens  and  a  miniature  incandescent  lamp.  The 
lens  focuses  the  interior  of  the  enclosed  furnace  (an  ideal  black 
body)  on  the  filament  of  the  lamp.  The  current  through 
the  filament  is  adjusted  until  the  tip  of  the  filament  is 
invisible  against  the  image  of  the  furnace.  When  this  is 
true,  both  must  be  emitting  similar  light,  and  therefore 
they  must  be  at  approximately  the  same  temperature.  A 
small  eye-piece  aids  in  observing  the  tip  of  the  filament. 
The  incandescent  lamp  filament  is  to  be  calibrated;  i.e.,  the 
current  necessary  to  heat  the  tip  of  the  filament  to  different 
temperatures  is  to  be  determined.  The  temperature  of  the 
filament  is  determined  by  finding  the  temperature  of  the 
furnace,  by  the  thermocouple,  when  the  two  have  the  same 
temperature.  (The  incandescent  lamp  circuit  contains  an 
ammeter  for  measuring  the  current,  which  is  omitted  from 

Fig.  31.) 

Observations. — While  the  furnace  is  slowly  heating,  and 
also  while  it  is  slowly  cooling,  observe  at  frequent  intervals, 


PYROMETRY.  117 

(A)  the  galvanometer  deflection  with  the  thermocouple, 
and  the  resistance  in  the  galvanometer  circuit;  (B)  the 
resistance  of  the  platinum  thermometer;  (C)  the  current 
through  the  filament.  The  three  observations  should  be 
made  in  succession  and  the  times  of  each  recorded.  (D) 
Calibrate  the  galvanometer  as  described  in  Exp.  LVIII,  if 
the  constant  is  not  furnished.  (E)  If  time  permits,  use  the 
optical  thermometer  to  determine  the  temperature  of 
various  distant,  brightly  heated  bodies;  e.  g.,  melted  silver, 
iron,  or  copper.  An  image  of  the  hot  body  is  formed  upon 
the  tip  of  the  filament,  and  the  current  through  the  latter 
is  adjusted  until  the  two  are  indistinguishable.  The  tem- 
perature corresponding  to  this  current  is  obtained  from  the 
calibration  (see  (e)  below). 

Report,  —  (a)  Tabulate  readings,  (b)  Plot  the  three  sets 
of  readings  against  the  time,  (c)  Make  a  second  plot  with 
resistance  as  abscissae  and,  for  ordinates,  the  platinum 
temperatures,  as  given  by  Callender's  equation, 

RI—RQ 

pi  —  IOO  -  . 


where  R0  is  the  extrapolated  resistance  at  o°  and  R1  is  the 
resistance  at  100°.  Transfer  to  this  plot  also  the  readings 
of  the  true  temperature,  /,  and  determine  the  mean  value  of 
Callender's  difference  constant,  8,  by  applying  at  several 
points  the  equation 


For  pure  platinum  8  is  1.50.  The  platinum  resistance 
thermometer  is  the  most  accurate,  convenient  method  of 
measuring  temperature  below  1000°. 

(d)  Construct  a  curve  which  gives  the  temperature  of  the 
tip  of  the  incandescent  lamp  filament  plotted  against  the 
current,  (e)  Finally,  determine  by  the  latter  curve  the 
temperatures  of  any  bodies  tested  with  the  optical  pyrometer 
and  record  the  results.  (For  a  discussion  of  the  error  in 


Ilg  HEAT. 

assuming  for  different  bodies  that  the  radiation  is  similar  to 
that  from  a  black  body,  see  the  above  reference  to  the 
Bulletin  of  the  Bureau  of  Standards  and  Haber,  "Thermo- 
dynamics of  Gas  Reactions"  pp.  281-291.) 

Questions. 

1.  Would  you   expect  the  platinum  resistance  thermometer  to 
attain  a  slightly  higher  or  a  slightly  lower   temperature   than  the 
thermocouple?     Explain. 

2.  Is  any  correction  required  for  the  absorption  of  the  lenses 
in  such  an  optical  pyrometer?     Explain. 


SOUND. 


/^s\ 


XXX.  THE  VELOCITY  OF  SOUND. 

Ttxt-book  of  Physics,  (Duff),  pp.  319,  320,  334-337;  Watson's  Physics, 
§§287,  288,  309;  Ames'  General  Physics,  pp.  337,  363;  Crew's 
General  Physics,  §213;  Poynting  and  Thomson,  Sound,  Chap.  VII- 

The  velocity  of  sound  in  a  medium  can  be  found  if  the 
wave-length,  L,  in  the  medium  of  a  note  of  frequency  n  can 
be  determined;  for  v  =  n  L.  If 
the  medium  be  contained  in  a 
tube,  one  end  of  which  is  closed, 
the  closed  end  must  be  a  node 
and  the  open  end  a  loop.  Hence 
the  length  of  the  tube  must  be 
an  odd  number  of  quarter  wave- 
lengths. Such  a  tube  will  reso- 
nate to  a  fork,  if  the  wave-length 
of  a  natural  vibration  of  the  pipe 
be  the  same  as  the  wave-length 
to  which  the  fork  gives  rise. 
Thus,  if  the  length  of  pipe  that 
resonates  to  a  fork  of  known 
pitch  be  measured,  we  have  the 
means  of  finding  the  velocity  of 
sound. 

A  long  glass  tube  is  mounted 
on  a  stand.  Water  is  introduced 
from  the  bottom,  where  is  at- 
tached a  rubber  tube  provided 
with  a  pinch-cock  and  connected 
to  a  glass  bottle.  By  raising  and 
lowering  the  bottle,  water  may  be 
brought  to  any  height  in  the  tube.  The  additional  connec- 
tions represented  in  figure  32,  permit  raising  or  lowering  the 
water  level  by  opening  pinch-cocks.  When  A  is  opened 

119 


FIG.  32. 


120  SOUND. 

the  tube  fills  and  the  tube  empties  upon  opening  D.  C 
and  B  are  pinch-cocks  which  regulate  the  rate  of  flow. 
The  vessels  are  so  large,  compared  with  the  capacity  of  the 
tube  that  only  rarely  is  it  necessary  to  transfer  water  from 
the  lower  to  the  upper  vessel. 

If  a  tuning-fork  be  vibrated  above  the  tube,  resonance  will 
first  take  place  when  the  air  column  is  approximately  one- 
quarter  wave-length  of  the  fork,  next  when  three-quarter 
wave-lengths,  etc.  In  reality,  the  first  loop  is  not  exactly  at 
the  open  end  of  the  pipe,  but  a  short  distance  beyond  the 
open  end.  The  distance  between  two  nodes  is  accurately 
half  a  wave-length,  and  it  is  from  this  distance  that  the 
wave-length  is  best  determined. 

The  tuning-fork  may  be  sounded  by  gently  striking  the 
end  of  the  fork  against  the  knee  or  a  block  of  soft  wood. 
The  fork  should  be  held  above  the  end  of  the  tube,  so  that 
the  plane  of  the  prongs  includes  the  axis  of  the  tube.  Each 
node  should  be  located  very  carefully,  at  least  four  times, 
each  location  being  tested  both  with  the  water  rising 
and  with  the  water  falling,  and  the  distance  of  each  position 
from  the  open  end  noted.  The  mean  is  taken  as  the  true 
distance.  The  whole  should  then  be  repeated  with  a  fork 
of  different  pitch.  Observe  the  temperature  and  barometric 
pressure,  and  measure  the  diameter  of  the  tube.  With  a 
little  practice  one  can  often  locate  nodes  corresponding  to 
the  higher  modes  of  vibration  of  the  fork.  This  should 
be  tried  and,  from  their  wave-lengths  and  the  velocity  of 
sound  as  already  determined,  the  pitch  of  these  higher  sounds 
can  be  calculated. 

The  pitch  of  the  forks  used  may  be  determined  by  com- 
parison with  a  standard  fork  by  the  method  of  beats.  The 
standard  is  mounted  on  a  resonance  box  and  is  set  in  vibra- 
tion by  pulling  the  prongs  together  with  the  fingers  and 
then  releasing  them.  The  fork  of  unknown  pitch  is  sounded 
in  the  usual  way,  and  the  end  of  the  shank  is  set  upon  the 
resonance  box  of  the  standard.  If  the  two  forks  are  of 
nearly  the  same  pitch,  beats  will  be  heard.  With  a  stop- 


THE    VELOCITY    OF    SOUND.  121 

watch  the  time  of  ten,  fifteen,  or  twenty  beats,  as  may  be 
convenient,  is  several  times  determined.  Dividing  by  the 
time,  we  have  the  number  of  beats  per  second,  and  this 
is  the  difference  of  pitch  of  the  two  forks.  To  determine 
which  fork  is  the  higher,  add  a  little  wax  to  one  prong  of 
the  fork  used  in  the  experiment.  Since  this  increases  the 
inertia  of  the  fork,  it  decreases  its  pitch.  If  originally  the 
two  forks  have  very  nearly  the  same  pitch,  so  that  there 
is  only  a  fraction  of  a  beat  per  second,  very  small  amounts 
of  wax  should  be  added.  A  large  piece  of  wax  might 
change  the  pitch  of  the  fork  from  above  that  of  the  standard 
to  below.  (Time  may  often  be  saved  by  comparing  the 
fork  with  the  standard  during  the  necessary  delays  of  the 
work  described  below.) 

The  velocity  of  sound  in  carbon  dioxide  may  be  deter- 
mined in  a  similar  manner.  The  water  surface  is  first 
lowered  to  the  bottom  of  the  tube  and  the  tube  is  filled  with 
the  gas  from  a  generator  through  a  small  tube  lowered  just 
to  the  water  surface  (not  below) .  The  generator  consists  of 
a  tube  filled  with  marble,  surrounded  by  dilute  hydrochloric 
acid.  In  filling  the  generator  with  marble,  use  only  whole 
pieces,  carefully  excluding  any  dust  or  pieces  small  enough 
to  drop  through  into  the  outer  vessel.  If  the  gas  is  not 
evolved  in  sufficient  quantity,  add  hydrochloric  acid  to  the 
outer  vessel.  When  the  air  in  the  tube  has  been  entirely 
displaced  by  the  gas,  a  lighted  match  introduced  into  the  top 
of  the  tube  will  be  extinguished.  The  delivery  tube  is  now 
withdrawn  from  the  resonance  tube,  while  the  gas  still  flows 
out  to  fill  the  volume  occupied  by  the  delivery  tube. 

The  water  surface  is  now  slowly  raised  and  the  nodes 
located  for  one  of  the  forks.  Observations  can  only  be  made 
with  the  water  rising,  for,  when  the  water  surface  is  lowered, 
air  enters  the  tube.  Hence  each  node  can  be  located  but 
once.  The  tube  is  again  filled  and  the  nodes  redetermined 
with  the  same  fork. 

From  the  distance  between  the  nodes  and  the  pitch  of  the 
fork  the  velocity  of  sound  is  determined. 


1.2  SOUND. 

For  each  gas,  find  the  correction  for  the  open  end;  that 
is,  the  displacement  of  the  loop  beyond  the  end  of  the  tube. 
Use  the  mean  position  of  the  highest  node  and  one-fourth 
of  the  mean  wave-length.  Find  what  fraction  this  displace- 
ment is  of  the  radius  of  the  tube. 

Calculate,  from  the  mean  values  of  the  velocities,  the 
velocity  of  sound  in  each  gas  at  o°  C.  (see  references). 
Find  also  the  ratio  of  the  specific  heats  from  the  velocity 
at  o°  C.,  the  standard  barometric  pressure  (both  in  absolute 
units)  and  the  density  as  given  in  Table  VI. 

Questions. 

1.  Explain  why  (a)  readings  are  made  with  both  rising  and  falling 
water  (b)  the  plane  of  the  prongs  of  the  fork  must  contain  the  axis 
of  the  tube. 

2.  What  is  the  influence  of  atmospheric  moisture  upon  the  velocity 
of  sound? 


XXXI.  VELOCITY  OF  SOUND  BY  KUNDT'S  METHOD. 

Text-book  of  Physics  (Duff),  p.  338;  Watson's  Physics,  §317;  Ames' 
General  Physics,  p.  364;  Crew's  General  Physics,  §215;  Poynting 
and  Thomson,  Sound,  pp.  115-117;  Watson's  Practical  Physics, 
§113- 

A  glass  tube,  A  G,  about  a  meter  long  and  about  3  cm. 
internal  diameter  is  closed  at  one  end  by  a  tight-fitting 
piston,  C,  and  at  the  other  end  by  a  cork  through  which 
passes  a  glass  tube  having  at  one  end  a  loosely  fitting  card- 
board disk,  D  (Fig.  33).  The  glass  tube  should  be  about  a 
meter  long.  A  little  dry  powdered  cork  is  sprinkled  in 
the  tube,  the  stopper  at  G  is  loosened,  and  a  current  of  air, 
dried  by  passage  through  several  drying  tubes,  is  slowly 
forced  through  the  hollow  rod  of  the  piston,  C.  The  stopper 
at  G  is  then  replaced  and  the  glass  tube,  F,  is  held  at  the 
center  and  stroked  longitudinally  with  a  damp  cloth.  The 
piston,  C,  is  adjusted  until  the  powder  collects  in  the  sharp- 
est attainable  ridges.  These  ridges  will  appear  where  the 
pressure  changes  are  least;  that  is,  at  the  loops.  Measure 
carefully  the  distance  between  two  extreme  ridges  and 


VELOCITY    OF    SOUND   BY   KUNDT's   METHOD.  123 

divide  by  the  number  of  segments  into  which  the  tube  is 
divided.  This  distance  (between  two  loops)  is  a  half  wave- 
length of  the  waves  in  the  tube.  Disturb  the  powder  and 
make  a  new  adjustment  of  the  piston,  C,  and  a  new  measure- 
ment of  the  half  wave-length.  Make  a  third  repetition  of 
the  adjustments  and  readings. 


—7^1 

1       1 

—  c  —  r~ 

j 

A        C 

D 

FIG.  33. 

6 

Fill  the  tube  with  another  dried  gas,  for  example,  carbon 
dioxide,  illuminating  gas,  hydrogen,  oxygen,  or  hydrogen 
sulphide,  and  determine  the  half  wave-length.  If  n  is  the 
constant  pitch  of  the  note  emitted  by  the  glass  tube  and  /  is 
the  wave-length  in  the  gas 

vi     li 
v=nl   .'.  —  =~i. 

^2        k 

Since  the  velocity  changes  at  the  same  rate  with  change  of 
temperature  in  all  gases  (see  references),  the  velocity  of 
sound  or  compressional  waves  at  zero  degrees  in  any  other 
gas  than  air  can  be  calculated  from  the  ratio  of  the  wave- 
lengths at  a  common  temperature,  and  the  velocity  in  air 
at  zero  degrees  (33,200  cm.  per  second). 

From  the  velocity  of  sound  at  zero  degrees  in  the  gases 
other  than  air  and  the  standard  atmospheric  pressure  cal- 
culate the  ratio  of  specific  heats,  7-  (see  references).  Table 
VI  gives  the  densities  of  the  more  common  gases  and  vapors 
at  zero  degrees  and  a  pressure  of  76  cm.  of  mercury  =  1013200 
dynes  per  square  centimeter. 

Questions. 

1.  Calculate    (a)    the   velocity  of   compressional  waves   in   glass, 
(b)  the  elasticity  E.      (Notice  that  each  end  of  the  glass  rod  must  be  a 
loop,  and  the  center  a  node.     The  density  of  glass  can  be  obtained 
from  Table  VIII.) 

2.  Why  must  the  glass  rod  be  set  in  longitudinal  vibration? 

3.  Why  does  the  powder  collect  at  the  loops? 


LIGHT. 


27.  Monochromatic  Light. 

The  simplest  and  most  useful  monochromatic  light  is  the 
sodium  flame.  Sodium  may  be  introduced  into  a  Bunsen 
flame  by  surrounding  the  tube  of  the  burner  with  a  tightly 
fitting  cylinder  of  asbestos  which  has  been  saturated  with  a 
strong  solution  of  common  salt  and  formed  into  cylindrical 
shape  by  wrapping  around  the  burner  while  still  damp. 
As  the  top  of  the  cylinder  is  exhausted,  it  should  be  torn  off 
and  the  rest  of  the  tube  pushed  up  into  the  lower  part  of 
the  flame.  A  piece  of  hard-glass  tubing  held  in  the  flame 
will  also  give  a  good  sodium  light. 

Elements  giving  red,  green,  blue,  and  violet  light  will  be 
found  in  Table  XVIII.  Salts  of  these  elements  (e.  g., 
KN03,  SrCl2,  CaCl2,  LiCl)  may  be  introduced  into  the  outer 
edge  of  a  bunsen  flame,  either  in  a  thin  platinum  spoon,  on 
copper  gauze,  or  by  a  piece  of  wood  charcoal  which  has 
absorbed  a  solution.  If  a  very  intense  light  is  not  required, 
a  vacuum  tube  is  a  very  satisfactory  source  (Table  XVIII). 
Intense  light  of  one  general  color  may  be  obtained  by  filtering 
sun  light  or  the  light  from  an  arc  light  through  colored  glass 
or  gelatine.  The  solutions  given  in  the  accompanying  table 
give  much  purer  monochromatic  light. 

Light  Filters  (Landolt).* 


Color 

Thickness 
of  layer 
(mm.) 

Aqueous 
solution  of 

j  Grams  per 

IOO  C.C. 

Average 
wave-length 
(Angstrom  units) 

Red.. 

20 

20 

,  Crystal  violet  560 
Potassium  chromate 

.005 
10. 

6560 

Green 

2O 
20 

;  Copper  chloride 
Potassium  chromate 

60. 
10. 

5330 

Blue.. 

20 
2O 

l  Crystal  violet 
1  Copper  sulphate 

.005 
15- 

4480 

*  Mann,  Manual  of  Advanced  Optics,  p.  185. 

124 


SPHERICAL    MIRRORS    AND    LENSES.  125 

28.  Rule  of  Signs  for  Spherical  Mirrors  and  Lenses. 

Mirrors. — Consider  the  side  upon  which  the  incident  light 
falls  as  the  positive  side  of  the  mirror.  If  the  object,  the 
image,  or  the  principal  focus  is  on  this  side,  their  respective 
distances,  (u,  v,  f=rj 2)  will  be  positive;  if  on  the  other  side, 
negative.  Therefore,  the  focal  length  (and  hence  radius)  is 
positive  for  concave  mirrors  and  negative  for  convex.  The 
object  distance,  u,  will  obviously,  in  most  cases,  be  positive. 

The  formula  for  all  spherical  mirrors  is : 

I        I        I        2 

U      V      f       Y 

if  the  signs  of  the  numerical  quantities  which  are  substituted 
for  u,  v,  f,  and  r  are  determined  by  the  above  rule. 

Lenses. — Let  all  the  distances,  u,  v,  f,  rlt  r2  be  positive 
for  the  double  convex  lens,  when  the  object  is  outside  the 
principal  focus;  that  is,  in  the  most  common  case.  The 
formula  for  all  lenses  is  then 


U       V       f 

As  an  illustration  of  the  application  of  this  rule,  consider  the 
signs  of  these  distances  when  an  image  of  a  real  object  is 
formed  by  a  double  concave  lens.  The  distance,  u,  of  the 
object  is  obviously  measured  on  the  same  side  as  it  would 
be  in  the  standard  case  of  the  double  convex  lens  and  is, 
therefore,  positive.  The  distances  /  and  v  are,  however 
measured  on  the  same  side  of  the  lens  as  the  object,  or 
opposite  to  the  standard  case  with  the  double  convex  lens, 
and  are,  therefore,  negative.  rlt  the  radius  of  the  front  face, 
is  on  the  same  side  as  the  object,  while  in  the  case  of  the 
double  convex  lens  this  radius  is  on  the  other  side,  therefore 
rlt  and  similarly  r2,  is  negative  for  a  double  concave  lens. 


XXXII.  PHOTOMETRY. 

Text-book  of  Physics  (Duff),  p.  353;  Watson's  Physics,  §§361-364; 
Ames'  General  Physics,  pp.  437,  442;  Watson's  Practical  Physics, 
pp.  382-387;  Edser,  Light,  pp.  9-20;  Stine's  Photometrical 
Measurements;  Palaz'  Photometry. 

The  intensity  of  illumination  of  a  surface  by  a  source  of 
light  of  small  area  varies  inversely  as  the  square  of  the 
distance.  Hence  it  follows  that,  if  two  lights  produce  equal 
intensities  of  illumination  at  a  point,  P,  their  illuminating 
powers,  or  the  intensities  of  illumination  they  can  produce  at 
unit  distances,  are  directly  as  the  squares  of  their  distances 
from  P.  This  is  the  basis  of  all  practical  methods  of  com- 
paring illuminating  powers. 

As  a  means  of  testing  when  two  different  sources  of  light 
produce  equal  illumination  at  a  point,  various  so-called 
screens  have  been  used.  The  one  that  has  been  most 
extensively  employed  is  Bunsen's  grease-spot  screen.  It  is 
based  on  the  fact  that  a  grease-spot  on  paper  is  invisible 
when  the  paper  is  equally  illuminated  on  both  sides,  since 
viewed  from  one  side  as  much  light  is  gained  by  transmission 
from  the  farther  side  as  is  lost  by  transmission  to  the  farther 
side. 

Another  screen  more  perfect  in  some  respects  is  that  of 
Lummer  and  Brodhun.  A  white  opaque  disk  (see  figure  34) 
is  illuminated  on  opposite  sides  by  the  two  sources  of  light. 
An  arrangement  of  mirrors  and  lenses  enables  one  eye  to 
view  both  sides  at  once.  Two  plane  mirrors  reflect  rays 
from  the  two  sides  into  a  double  glass  prism.  This  consists 
of  two  separate  right-angled  prisms,  the  largest  face  of  one 
being  partly  beveled  away  and  the  two  being  cemented  to- 
gether by  Canada  balsam,  which  has  the  same  optical  density 
as  the  glass,  and  therefore  reflects  no  light.  The  central 
rays  from  the  left  pass  through  the  double  prism  to  the  tele-. 

126 


PHOTOMETRY.  127 

scope  while  the  marginal  rays  are  totally  reflected  by  the 
beveled  edge.  The  marginal  rays  from  the  right  are  totally 
reflected  and  reach  the  telescope,  but  the  central  rays  pass 
through.  Thus  the  eye  sees  a  circular  portion  of  the  left 
side  of  the  opaque  disk  and  a  surrounding  rim  of  the  right 
side. 

To  eliminate  error  from  lack  of  symmetry,  the  lamps 
compared  should  be  interchanged  in  the  course  of  the  readings 
or  the  screen  should  be  rotated  180°. 

The  lights  to  be  compared  are 
mounted  at  opposite  ends  of  a 
graduated  bar  3  meters  long,  which, 
with  a  parallel  bar  and  suitable  sup- 
ports, constitutes  the  photometer 
bench.  The  screen  is  mounted  on  a 
carriage  movable  along  the  bench. 

Many  light  standards  have  been 
employed.  A  candle  of  certain  care-  FlG 

fully  specified  dimensions  was  long 
employed,  and  the  illuminating  power  of  such  a  candle  is  still 
regarded  as  the  unit  and  called  "one  candle-power,"  but, 
for  practical  purposes  in  testing,  some  other  standard  is 
usually  employed.  The  best  such  standard  is  a  lamp,  with 
a  wick  of  specified  form  and  dimensions,  burning  amyl 
acetate  with  a  flame  of  specified  height.  (See  references.) 
Its  relation  to  the  "candle-power"  is  i  c.  p.  =  1.14  amyl 
acetate  units.  For  most  purposes  an  incandescent  lamp 
that  has  been  standardized  is  the  most  useful  standard, 
especially  in  the  study  of  incandescent  lamps;  but  it  must 
not  be  used  any  great  length  of  time  without  being  re-stand- 
ardized, since  its  illuminating  power  changes  with  prolonged 
use. 

The  chief  difficulty  in  comparing  two  different  forms  of 
light  is  due  to  the  fact  that  a  difference  of  quality  of  the 
two  lights  renders  perfectly  equal  and  similar  illumination 
of  the  two  sides  of  the  screen  impossible.  This  difficulty 
is  still  more  marked  in  the  study  of  arc-lights  (for  mechan- 


128  LIGHT. 

ical  arrangements  see  Stine,  p.  236),  for  which  it  is  best  to 
use  as  an  intermediate  unit  a  very  powerful  incandescent 
lamp.  The  latter  may  be  standardized  by  comparison  with 
an  ordinary  incandescent  lamp,  which  again  is  compared 
with  an  amyl  acetate  standard.  Before  connecting  a  lamp 
to  a  circuit,  ascertain  that  the  voltage  is  not  excessive  for  that 
particular  lamp. 

(A)  Carefully     standardize     an    incandescent     lamp,     for 
use  as  a  working  standard,  by  comparison  with  either  an 
amyl  acetate  lamp  or  a  standardized  incandescent  lamp. 
If  the  latter  is  used,  the  lamps  should  be  in  parallel,  that  the 
voltage  may  be  the  same,  and  a  variable  resistance  should 
also  be  in  the  circuit,  by  varying  which  the  voltage  across 
the   lamps  is  maintained  at  the  value  prescribed  for  the 
standard.     See  that  the  filament  of  the  standard  is  in  the 
marked  azimuth  and  note  the  position  of  the  filament  for 
which  the  other  lamp  is  standardized. 

In  each  case  several  settings  of  the  screen  should  be 
rapidly  made,  and  then  the  screen  reversed  and  several  more 
made.  The  calculations  may  be  facilitated  by  Table  XXL 

(B)  The  law  of  inverse  squares  should  be  tested  by  com- 
paring two  somewhat  different  incandescent  lamps  (i)  when 
3  m.  apart,   (2)  when  2.5m.  apart,   (3)  when  2  m.  apart 
on  the  photometer  bench.     The  ratio  of  their  illuminating 
powers,  as  deduced  in  the  three  cases,  should  be  a  constant. 

(C)  The    horizontal   distribution   of   candle-power    about 
an  incandescent  lamp  should  be  studied.     This  incandes- 
cent lamp  should  be  connected  in  parallel  with  the  working 
standard  and  the  voltage  maintained  at  the  value  for  which 
the  latter  was  standardized.     The  lamp  should  be  mounted 
on  the  revolving  lamp-holder  of  the  photometer,  care  being 
taken  to  have  the  center  of  the  filament  at  the  same  height 
as  the  center  of  the  screen.     The  lamp  is  first  turned  to  the 
standard  position,  i.  e.,  the  position  in  which  the  plane  of 
the  shanks  of  the  filament  is  at  right  angles  to  the  photom- 
eter bench,   and   a  marked  face  of  the  lamp  is  toward  the 
screen.     The   candle-power  of  the  lamp  is  to  be  found  in 


PHOTOMETRY.  1 29 

this  position  and  at  positions  30°  apart  as  the  lamp  is  ro- 
tated through  360°.  Two  careful  readings  should  be  made 
at  each  angle  and  the  c.  p.  deduced  from  the  mean.  The 
mean  of  all  these  values  of  the  c.  p.  gives  the  mean  horizon- 
tal candle-power,  A  curve  should  be  plotted,  giving  the 
distribution  of  c.  p.  in  polar  co-ordinates.  The  mean  hori- 
zontal candle-power  is  more  easily  determined  by  continu- 
ously rotating  the  lamp  about  a  vertical  axis  by  means  of  a 
small  motor. 

(D)  Efficiency   of  an   Incandescent   Lamp. — Keeping  the 
potential  of  the  working  standard  at  the  proper  point  (or 
calibrating  and  using  a  lamp  which  may  be  connected  to  the 
lighting  circuit  if  the  potential  of  that  is  constant),  apply 
various  potentials  to  the  lamp  used  in  (C)  at  intervals  be- 
tween about  25%  below  the  normal  voltage  to  25%  above. 
For  each  potential,  determine  the  candle-power  and  current. 
Calculate  the  watts   consumed   and  the  watts  per  candle- 
power. 

In  the  report,  plot  in  three  curves,  with  volts  as  abscissae, 
(a)  current,  (b)  candle-power,  (c)  watts  per  candle-power. 
The  scales  of  the  three  curves  should  be  shown  on  the 
vertical  axis. 

(E)  Mean  Spherical  Candle-power. — With  the  lamp  in  the  standard 
position  of  (C),  find  the  c.  p.  at  intervals  of  30°  in  a  vertical  circle  by 
rotating  the  lamp  about  a  horizontal  axis.     After  this,  start  again 
from  the  standard  position  and  first  turn  the  lamp  through  45°  in 
azimuth  (or  around  a  vertical  axis),  and  then,  as  before,  find  the  c.  p. 
at  intervals  of  30°  in  the  vertical  circle  of  45°  azimuth,  and  so  for  the 
vertical  circles  of  90°  and  135°  azimuth.     As  before,  plot  the  curves 
of  distribution  in  polar  co-ordinates. 

To  find  the  mean  spherical  candle-power  omit  any  repetitions 
and  take  the  mean  of  the  readings  in  the  following  positions : 

1 .  At  tip      i 

2.  At  60°,  120°,  240°,  300°  on  the  vertical  circles  of  o°  and  90° 

azimuth 8 

3.  At  30°,  150°,  210°,  330°  on  the  vertical  circles  of  o°,  45°,  90°, 

135°  azimuth    16 

4.  12  equidistant  positions  on  horizontal  circle      12 

5.  At  base  (o)     i 

Total, 38 

These  directions  are  chosen  because  they  are  nearly  uniformly 
distributed  in  space. 

9 


130  LIGHT. 

(F)  If  time  permit,  study  the  differences  of  quality  of 
light  given  by  different  sources;  e.  g.,  compare  an  oil  lamp  and 
an  incandescent  lamp  using  interposed  colored  glasses: 
(i)  a  pair  of  red  glasses,  (2)  of  yellow  glasses,  (3)  of  blue 
glasses.  Calculate  the  relative  illuminating  powers  in  each 
case. 

The  possible  error  may  be  deduced  as  usual  from  the 
mean  deviation  of  the  readings  in  a  set. 

Questions. 

1.  Explain  the   deviation   of  the   current-voltage   curve   from   a 
straight  line. 

2.  What  is  the  advantage  in  increasing  the  voltage  applied  to  an 
incandescent  lamp?     Disadvantage? 


XXXIII.  SPECTROMETER  MEASUREMENTS. 

Text-book  of  Physics  (Duff),  pp.  387,  436,  437,  440;   Watson's  Phys- 
ics,  pp.   468,  469,   493;    Antes'  General  Physics,   pp.   459,   460, 
505,  506;  Crew's  General  Physics,  p.  510;  Edser's  Light,  pp.  86-91. 

A  spectrometer  consists  of  a  framework  supporting  a 
telescope  and  a  collimator,  both  movable  about  a  vertical 
axis,  and  a  platform  movable  about  the  same  axis.  The 
platform  is  for  supporting  a  prism  or  grating.  The  colli- 
mator is  a  tube  containing  an  adjustable  slit  at  one  end  and 
a  lens  at  the  other  end.  The  purpose  of  the  collimator  is 
to  render  light  coming  from  the  slit  parallel  after  it  leaves 
the  lens.  (Only  when  the  light  that  falls  on  a  prism  is  par- 
allel light,  that  is,  light  with  plane  wave  front,  does  it  seem 
when  emerging  from  the  prism  to  come  from  a  clearly  de- 
fined source.  When  it  is  not  parallel,  there  is  spherical 
aberration.)  Hence  the  slit  of  the  collimator  should  be  in 
the  principal  focus  of  the  lens.  The  telescope  is  for  the 
purpose  of  viewing  the  light  that  comes  from  the  collimator, 
either  directly  or  after  the  light  has  been  refracted  or 
reflected.  Hence,  since  the  light  that  comes  from  the  colli- 
mator is  supposed  to  be  parallel,  that  is,  as  if  it  came  from 
a  very  distant  source,  it  follows  that  if  the  telescope  is  to 


SPECTROMETER    MEASUREMENTS.  131 

receive  the  light  and  form  a  distinct  image  of  the  slit,  the 
telescope  must  be  focused  as  for  a  very  distant  object  (theo- 
retically an  infinitely  distant  one) . 

The  first  adjustment  is  to  focus  the  telescope.  First 
focus  the  eye-piece  of  the  telescope  on  the  cross-hairs  and 
then  focus  the  whole  telescope  on  a  distant  object  out  of 
doors.  The  telescope  will  now  be  in  focus  for  parallel  rays. 
Turn  the  telescope  to  view  the  image  of  the  slit  formed  by 
the  collimator  and  adjust  the  slit  until  its  image  is  seen 
most  distinctly. 

That  the  instrument  should  be  in  complete  adjustment, 
it  is  necessary  that  the  telescope,  collimator,  and  platform 
should  rotate  about  the  same  axis,  and  that  the  optical  axes 
of  the  telescope  and  the  collimator  should  be  perpendicular 
to  this  axis  of  rotation.  For  fine  work  spectrometers  are 
made  with  all  these  parts  separately  adjustable,  but  simpler 
instruments  have  the  telescope  and  collimator  put  into  per- 
manent adjustment  by  the  instrument  maker.  In  any  case 
the  telescope  and  collimator  should  not  be  adjusted  for  level 
without  the  advice  of  an  instructor. 

Adjustment  of  Prism. — The  refracting  edge  of  the  prism 
must  be  made  parallel  to  the  axis  of  the  instrument.  Place 
the  prism  on  the  platform  with  one  of  the  faces  perpendicular 
to  the  line  joining  two  of  the  leveling  screws.  Turn  the 
collimator  slit  horizontal  and  place  the  telescope  so  as  to 
receive  the  image  of  the  slit  reflected  from  this  face  of  the 
prism.  Adjust  these  two  leveling  screws  until  the  image  of 
the  stationary  edge  of  the  slit  coincides  with  the  horizontal 
cross-hair.  Then  observe  the  image  reflected  from  the 
other  face  and  adjust  the  third  leveling  screw  until  the  edge 
of  this  image  is  on  the  horizontal  cross-hair.  A  little  con- 
sideration will  show  that,  when  these  two  adjustments  have 
been  made,  both  faces,  and  therefore  the  refracting  edge, 
are  parallel  to  the  axis.  Restore  the  collimator  slit  to  the 
vertical  position. 

Measurement  of  the  Angle  of  a  Prism. — Method  (A). — The 
prism  should  be  so  placed  that  the  faces  are  about  equally 


132  LIGHT. 

inclined  to  the  collimator.  To  secure  good  illumination,  the 
edge  of  the  prism  should  be  near  the  axis  of  the  instrument. 
The  telescope  is  turned  to  view  the  image  of  the  slit  in  the 
two  faces  alternately,  and  the  scale  and  vernier  read  when 
the  slit  and  cross-hair  coincide,  the  slit  being  narrowed  until 
barely  visible.  If  the  scale  is  provided  with  two  verniers, 
to  eliminate  error  from  eccentricity,  always  read  them  both. 
Half  of  the  angle  between  the  two  positions  of  the  telescope 
gives  the  angle,  A,  of  the  prism,  as  may  be  readily  seen  by 
drawing  a  diagram.  The  readings  on  each  side  should  be 
repeated  three  times. 

Method  (B). — The  following  method,  which  is  some- 
times easier  than  the  preceding,  may  be  used  if  the  platform 
that  carries  the  prism  can  be  rotated  and  the  rotation  read 
by  a  scale.  Turn  one  face  of  the  prism  so  as  to  reflect  the 
image  of  the  slit  into  the  telescope.  Adjust  the  telescope 
until  the  vertical  cross-hair  coincides  with  the  slit  and  then 
read  the  platform  scale.  Now  rotate  the  platform  until  the 
other  face  of  the  prism  reflects  the  slit  and  again  read  the 
platform  scale.  The  difference  of  the  readings  is  1 80  ±  A,  as 
may  readily  be  seen  by  drawing  a  figure.  The  observation 
should  be  repeated  at  least  three  times. 

We  are  now  in  a  position  to  make  a  final  measurement 
for  finding  the  index  of  refraction  of  the  glass  of  the  prism 
for  any  particular  light  of  the  spectrum,  for  instance, 
sodium  light  (see  p.  124).  The  only  additional  measure- 
ment necessary  is  the  deviation  produced  by  the  prism  when 
it  is  in  such  a  position  that  it  gives  a  minimum  deviation  to 
the  light  refracted  through  it. 

Minimum  Deviation. — The  position  of  minimum  devia- 
tion is  such  that  the  image  of  the  slit  seen  in  the  telescope 
moves  in  the  same  direction  (that  of  increasing  deviation) 
no  matter  which  way  the  platform  carrying  the  prism  is 
turned.  There  are,  of  course,  two  positions  in  which  the 
deviation  can  be  obtained,  one  with  the  refracting  edge  turned 
toward  the  right  of  the  observer,  and  the  other  with  it 
toward  the  left.  The  deviation  in  each  case  is  the  angle 


SPECTROMETER    MEASUREMENTS.  133 

between  the  corresponding  position  of  the  telescope  and  its 
position  when  looking  directly  into  the  collimator,  the  prism 
being  removed.  But  it  is  not  necessary  to  remove  the 
prism,  for  it  is  easily  seen  that  the  minimum  deviation  must 
also  be  equal  to  half  of  the  angle  between  the  two  positions 
of  the  telescope  when  observing  the  minimum  deviation. 


FIG.  35. 

These  two  positions  should  be  observed  three  times  success- 
ively, and  the  mean  value  for  the  minimum  deviation,  D, 
taken.  From  A  and  D  the  index  of  refraction  may  be  cal- 
culated by  the  formula 

sin 


If  time  permit,  determine  the  index  of  refraction  for  as 
many  other  wave-lengths  (colors)  as  possible  (see  p.  124). 

The  possible  error  of  the  determination  of  the  refractive 
index  can  be  calculated  by  means  of  formulae  deduced  by 
the  calculus,  as  explained  on  pp.  7,  8.  A  simple,  but 
less  accurate  method  is  to  recalculate  n  with  A  and  D, 
increased  by  their  mean  deviations  and  to  consider  the 
difference  between  this  value  and  the  original  value  as  the 
possible  error. 

It  is  probable  that  in  this  experiment  there  are  other 
sources  of  error  that  exceed  mere  error  in  reading  the 
scale;  e.  g.,  (i)  The  faces  of  the  prism  may  not  be  true 
planes,  (2)  the  divided  circle  may  not  be  uniform,  (3)  the 
center  of  the  circular  scale  may  not  coincide  with  the  cen- 
ter of  the  instrument,  (4)  the  various  adjustments  may  not 
be  perfect,  (5)  there  may  be  difficulty  in  fixing  the  position 


134  LIGHT. 

of  minimum  deviation.  These  errors  might  be  eliminated  by 
repeating  all  the  adjustments  and  observations  many  times 
and  using  different  parts  of  the  divided  scale.  There  is  no 
other  way  of  allowing  for  them. 

Questions. 

1.  Give  both  physical  and  mathematical  definitions  of  the  refract- 
ive index. 

2.  Why  is  monochromatic  light  used? 

3.  Why  is  the  minimum  deviation  chosen? 


XXXIV.  MEASUREMENT  OF  RADIUS  OF  CURVATURE. 

Glasebrook  and  Shaw,  Practical  Physics,  pp.  339-343;  Edser,  Light, 
pp.  116-121;  Koklrausch,  pp.  174-176. 

The  radius  of  curvature  of  a  surface  may  be  determined 
from  the  size  or  position  of  the  image  which  the  spherical 
surface,  regarded  as  a  mirror,  forms  of  a  definite  object. 
Method  (A)  below  is  especially  applicable  to  the  measure- 
ment of  the  radius  of  curvature  of  convex  surfaces,  and 
method  (B)  to  concave  surfaces. 

(A)  Two  bright  objects  (see  Fig.  36)  are  placed  on  a  line 
at  right  angles  to  the  axis  of  the  spherical  surface,  the 
intersection  of  the  line  and  the  axis  being  at  a  considerable 
distance  A,  from  the  surface,  and  each  object  being  at  a 
distance  L/  2  from  the  axis.  If  the  apparent  distance  be- 
tween the  images  of  the  two  objects  be  /,  the  radius  of  curva- 
ture of  the  surface  is 


~ 

the  +  sign  being  used  for  a  concave  surface  and  the  —  sign 
for  a  convex. 

Proof. 

(For  convex  mirror.) 

Let  <i  =  true  distance  between  the  images,  x  =  distance  of  images 
from  mirror. 
By  geometry 

L    A+r     d_A+x       .  L     (A+r)(A+x) 
d~r-x'    l~    A    '     •''  I  "'    (r-x)A      ' 


MEASUREMENT   OF   RADIUS    OF   CURVATURE.  135 

By  the  equation  for  spherical  mirrors  (see  p  125). 
\  112 


x     r     A     r 

r  —  x_A+r 

roc  ~  Ar 

L    A  +  x  A 

.'.    -y  =  --  =1    +  —  . 
/  X  X 

From  (  i  )  —  =  i  H  — 

OC  T 

L  iA 


The  radius  of  curvature  should  be  found  for  both  surfaces 
of  a  double  convex  lens.  The  lens,  preferably  the  one  used 
in  Exp.  XXXV,  if  that  has  already  been  performed, 
is  fitted  in  a  clamp  in  a  darkened  recess.  At  some  distance 
are  two  vertical  slits  illuminated  from  behind  by  incan- 
descent lamps  (or  the  lamp  filaments  themselves  may  be 
used)  and  between  them  a  *> 
telescope.  The  telescope  and 
the  lens  are  adjusted  until,  on 
looking  through  the  telescope 
toward  the  lens,  the  illumi- 
nated slits  are  seen  reflected 
from  the  near  surface  of  the 
lens.  Distinguish  these  im- 

ages from  the  images  produced  by  the  rear  surface  of  the  lens 
by  the  change  of  focus  necessary  to  make  one  pair  of  images 
most  distinct,  and  then  to  make  the  other  pair  most  distinct, 
or,  by  observing  the  two  images  of  a  light  held  just  outside 
one  of  the  slits.  Remember  that  the  telescope  inverts. 
A  paper  scale  is  pinned  over  the  lens  so  that  the  upper  edge 
is  just  below  the  center  of  the  lens.  The  telescope  is  focused 
upon  the  scale,  and  rotated  until  the  vertical  cross-hair 
bisects  one  of  the  images  from  the  near  surf  ace  'of  the  lens, 
and  the  scale  read  where  crossed  by  the  cross-hair.  (Esti- 


136  LIGHT. 

mate  tenths  of  millimeters  as  always.)  A  similar  reading 
is  made  for  the  other  image.  The  difference  between  the 
two  readings  gives  the  apparent  distance,  /,  between  the 
images.  At  least  six  independent  determinations  of  this 
distance  should  be  made.  Measure  the  distance,  L,  between 
the  slits,  the  distance,  A,  from  the  lens  surface  to  the  line 
joining  the  slits  and  substitute  in  the  formula. 

From  the  two  radii  of  curvature  and  the  focal  length,  if 
known,  calculate  the  refractive  index,  n,  of  the  glass  of  the 
lens  by  means  of  the  formula 


(B)  As  a  concave  mirror  we  may  use  one  of  the  surfaces 
of  a  concave  lens,  mounted  in  a  lens-holder.  To  reduce 
reflection  from  the  other  surface,  the  latter  may  be  covered 
by  moist  filter  paper.  The  radius  is  determined  from  u, 
the  distance  of  the  object,  v,  the  distance  of  the  image  and 
the  formula 

112 

-+-=-,  (seep.  125). 

u     v     r 

In  locating  the  image,  use  is  made  of  the  fact  that  if  the 
eye  is  a  considerable  distance  off,  a  real  image  can  be  seen 
in  space  as  well  as  a  virtual  image,  and  a  wire,  needle,  or 
pointer  is  moved  about  until  there  is  no  parallax  between  it 
and  the  image;  i.  e.,  until,  when  the  eye  is  moved  about, 
there  is  no  relative  motion  of  the  two. 

A  vertical  wire  illuminated  by  a  lamp,  behind  which  is 
a  sheet  of  white  paper,  is  a  convenient  object,  and  a  second 
mounted  wire  is  moved  about  until  it  coincides  with  the 
image  of  the  first  (see  (B)  Exp.  XXXV).  The  image 
should  be  found  for  at  least  the  following  three  typical 
positions  of  the  object.  For  each  position  make  several 
settings  and  from  the  means  determine  u  and  v,  and  from 
them  determine  r. 

(i)  Let  the  object  be  at  a  considerable  distance  from  the 
mirror. 


FOCAL    LENGTH    OF    LENS.  137 

(2)  Let  the  object  be  at  the  center  of  curvature  of  the 
mirror.     In  this  position  the  image  and  the  object  coincide. 

(3)  Let  the  object  be  within  the  principal  focus.     For  this 
position  the  wire  locating  the  image  must  be  on  the  other 
side  of  the  lens.     This  wire  is  moved  about  until  the  pro- 
longation above  the  lens  of  the  image  of  the 

first   wire  coincides  with  what  is  seen  of  the 
second  wire  above  the  lens. 

In  the  report,  sketch  the  relative  positions 
of  mirror,  image,  and  object,  and  state  whether    ' 


the  image  was  magnified  or  diminished,  erect 
or  inverted. 

(C)  If  time  permit,  check  your  results  with  a 
spherometer  (see  p.  16).  The  spherometer 
should  be  read  alternately  on  a  plane  surface 
and  on  the  lens.  Let  a  =  difference  in  the 
two  readings  (see  Fig.  37)  and  r  =  radius  of  the  circle  of  the 
legs.  Then  the  radius  of  curvature  of  the  lens  is 


as  may  be  easily  shown. 

Questions. 

1.  For  what  lenses  would  the  first  method  of  determining  the  radius 
of  curvature  be  preferable,   and   when  would  the  spherometer  be 
preferable  ? 

2.  What  objection  is  there  to  determining  the  radius  of  curvature 
of  the  farther  face  of  a  convex  lens,  considering  it  a  concave  surface? 

3.  What  advantages  has  the  method  used  in  (B)  for  locating  real 
images  over  the  use  of  a  screen  ? 

4.  How  could  you  directly  determine  with  a  screen  the  center  of 
curvature  of  a  concave  mirror? 


XXXV.  FOCAL  LENGTH  OF  A  LENS. 

Text-book  of  Physics  (Duff),  pp.  392-398;  Watson's  Physics,  pp.  471- 
479;  Ames'  General  Physics,  pp.  470—483;  Crew's  General  Physics, 
pp.  466-469;  Edser,  Light,  pp.  110-116;  Glazebrook  and  Shaw, 
Practical  Physics,  pp.  343-352. 

The  focal  length  of  a  lens  is  the  distance  from  the  optical 
center  of  the  lens  to  the  focus  for  rays  of  light  from  an 


138  LIGHT. 

infinite  distance;  i.  e.,  for  plane  waves.  If  /  is  the  focal 
length,  u  the  distance  of  the  object  from  the  lens,  and  v 
that  of  the  image,  then,  with  the  convention  respecting  signs 
given  on  page  125,  for  all  lenses 

iii 

v     u     f 

(A)  Real  Image.— (li  Exp.  XXXIV  has  preceded,  use 
the  same  lenses.)  An  "object,"  the  lens,  and  a  screen 
for  receiving  the  image  of  the  object,  are  mounted  so  that 
they  can  be  moved  along  a  graduated  scale.  A  convenient 
form  for  the  object  is  a  wire  cross  or  gauze,  mounted  in  a 
black  wooden  support,  and  illuminated  from  behind  by  an 
incandescent  lamp.  The  lens  is  clamped  in  a  wooden  frame 
movable  along  the  scale.  This  should  grasp  the  lens  on  the 
sides,  leaving  the  top  and  bottom  clear.  The  distance  from 
the  center  of  the  lens  to  some  point  on  the  support  must  be 
determined  once  for  all  and  applied  as  a  correction  to  the 
readings.  With  object  and  screen  in  fixed  positions  that  are 
recorded,  the  lens  is  adjusted  until  the  image 
on  the  screen  is  as  distinct  as  possible  and 
its  position  is  then  recorded.  This  should 
be  done  several  times,  and  the  mean  taken 
for  the  position  of  the  lens.  Keeping  object 
and  screen  fixed  and  moving  the  lens  about, 
,  another  image  will  be  found,  for  which  similar 

observations  should  be  made.  Calculate  /  from  the  averages 
of  all  the  values  of  u  and  v.  The  object  and  screen  should 
then  be  shifted  and  the  observations  repeated.  From  the 
two  sets  of  observations  a  mean  value  of  /  is  deduced. 

Study  of  Spherical  Aberration. — Determine  /  for  the  central 
part  of  the  lens  by  covering,  with  a  pasteboard  screen, 
all  but  a  central  disk  of  about  one-third  the  diameter  of 
the  lens.  Similarly  determine  /  for  the  edge  of  the  lens, 
using  a  diaphragm  covering  all  but  the  edge. 

Study  of  Chromatic  Aberration. — Using  the  entire  lens, 


FOCAL    LENGTH    OF    LENS.  139 

determine  /  for  red  light  by  placing  red  glass  before  the  lens 
or  object,  and  similarly  for  blue  or  green  light. 

In  the  report,  tabulate,  for  comparison,  the  different 
mean  values  of  /. 

(B)  Virtual  Image.  —  In  the  preceding  a  real  image  was 
observed,  but  the  focal  length  may  also  be  found  from 
observations  of  a  virtual  image.  The  following  directions 
apply  to  a  divergent  lens.  A  vertical  dark  line  on  a  white 
background  serves  as  object.  The  image  (between  the  lens 
and  the  object)  is  located  with  a  short  vertical  wire,  which 
is  moved  back  and  forth  until  a  position  is  found  where 
the  image  of  the  dark  line  seen  through  the  lens  (a  in  Fig.  38) 
appears  at  the  same  distance  as  the  portion  of  the  wire  seen 
just  below  or  above  the  lens  (b  in  Fig.  38).  This  is  secured 
when  there  is  no  relative  motion  of  the  image  and  this  wire 
as  the  eye  is  moved  horizontally,  i.  e.,  the  wire  appears  as 
the  prolongation  of  the.  image  of  the  dark  line  or  remains 
equidistant  from  such  a  prolongation,  v  will  be  the  distance 
from  the  center  of  the  lens  to  this  wire  which  locates  the 
image.  Using  a  longer  wire  as  the  object  and  the  dark  line 
to  locate  the  image,  this  method  may  be  applied  to  the 
virtual  image  of  a  convergent  lens. 

Estimate  the  possible  error  of  a  typical  measurement 
of  /.  .  Since  practically  all  the  error  is  in  the  location  of  the 
lens,  the  distance  between  the  object  and  the  screen  may 
be  considered  free  from  error.  If  this  distance  is  desig- 
nated by  w  ,  the  formula  becomes 


, 

u     w—u    j 

from  which  a  formula  may  easily  be  derived  for  the  possible 
error  in  /  in  terms  of  the  possible  error  in  u.  The  latter  may 
be  taken  as  the  mean  deviation  from  the  mean  in  the  location 
of  the  lens  (see  p.  4). 

If  Exp.  XXXIV  has  preceded,  determine  the  refract- 
ive index  of  the  glass  from  the  focal  length  and  the  radii 
of  curvature. 


140  LIGHT. 

Questions. 

1.  What  is  the  minimum  distance  between  object  and  screen  to 
secure  a  real  image?     The  maximum  distance  between  object  and 
lens  to  secure  a  virtual  image? 

2.  What  advantage  is  there  in  covering  with  a  diaphragm  all  but 
the  central  portion  of  a  lens?     What  disadvantage? 

3.  What  is  the  cause  of  chromatic  aberration? 

4.  What  sort  of  a  lens  would  show  large  spherical   aberration? 
Large  chromatic  aberration? 


XXXVI.  LENS  COMBINATIONS. 

Edser,  Light,  Chaps.  VI,  VII,  X;  Watson's  Practical  Physics,  pp.  358- 
367;  Drude's  Optics,  pp.  44-46,  66-72;  Hastings'  Light,  Appendix 
A.  Antes'  General  Physics,  pp.  488,  493,  494. 

(A)  Determination  of  Principal  Foci.  Calculation  from 
Focal  Lengths  and  Separation.  —  Let  two  lenses  of  focal  lengths, 
/!,  and/2,  be  separated  a  distance  d.  An  object  at  a  distance 
u  from  the  first  lens  forms  an  image  at  a  distance  v  deter- 
mined by  the  equation 


v    ^     u       ufl  ' 

This  image  acts  as  an  object  for  the  second  lens  at  a 
distance  d  —  v.  Hence  the  distance  of  the  final  image  from 
the  optical  center  of  the  second  lens  is  given  by  the  equation 

i       i         i         i  u~fi 

v'    /a     d~v    /2     du-dfi-ufi 

If  u  is  infinite,  vr  is  the  distance,  V,  of  the  principal  focus 
from  the  optical  center  of  the  second  lens, 


Experimental  Location.  —  Determine  experimentally  the 
position  of  this  principal  focus  by  finding  the  position  in 
which  an  object  must  be  placed  for  clear  vision  when  it  is 
viewed  through  the  combination  with  a  telescope  focused 
for  a  very  distant  object.  Compare  with  the  calculated 
position.  Determine  similarly  the  other  principal  focus. 


LENS    COMBINATIONS.  141 

(B)  Determination  of  Focal  Length.  —  To  determine  the 
focal  length,  the  position  of  the  "principal  points"  must 
be  known,  as  well  as  the  principal  foci.  With  thin  lenses, 
the  principal  points  practically  coincide  at  the  so-called 
"optical  center."  In  thick  lenses  or  lens  combinations  they 
may  be  considerably  separated.  The  focal  length  is  the 
distance  from  either  principal  focus  to  the  nearer  principal 
point.  The  equations  for  locating  the  image, 

iii 
-+-=7, 
it    v    f 

and  for  finding  the  linear  magnification, 

7' 

M  =  - 

u 

are  applicable  in  all  cases,  if  u  and  v  are  measured  from  the 
principal  points. 

Since  it  is  difficult  to  locate  the  principal  points,  a  method 
is  often  employed  for  determining  the  focal  length  which 
eliminates  their  position.  Suppose  that  the  linear  mag- 
nification is  Mlt  when  the  distance  of  the  object  from  the 
principal  plane  is  ult  and  M2  when  the  object  is  moved 
until  its  distance  is  u. 


If  the  focal  length  is  small,  the  magnification  should 
be  determined  with  a  micrometer  microscope.  A  carefully 
graduated  scale  is  a  convenient  object  and  the  size  of  the 
image  of  one  or  more  divisions  is  measured  for  two  positions 
of  the  object  a  known  distance  apart.  (Principle  of  Abbe's 
Focometer.) 

Thus  determine  the  focal  length  of  the  combination 
used  in  (A).  Determine  also  the  focal  length  of  both  the 
objective  and  the  eye-piece  of  a  telescope,  using  the  same 


142  LIGHT. 

one  as  employed  in  Exp.  XXXVII,  if  that  has  pre- 
ceded, and  calculate  the  magnifying  power  for  great 
distances. 

In  the  report,  draw  careful  figures  of  the  lens  combina- 
tions, representing  the  principal  foci  and  principal  points 
as  calculated  from  the  final  mean  results. 

(C)  If  the  principal  points  of  a  convergent  system  are 
close  together,  i.  e.,  if  the  lens  or  lenses  may  be  said  to  have 
an  optical  center,  we  may  use  the  following  approximate 
method:  If  w  is  the  distance  between  object  and  image, 
and  x  that  between  the  two  positions  of  the  lens  for  real 
images,  u=(w±x)/2,  v=(w±x)/2.  Substituting  these 
values  in  the  formula  we  get 

Zf2  —  X2 

4W 

If  time  permit,  try  this  method  for  the  combination  of 
lenses. 

Describe  a  lens  combination  which  (i)  magnifies  without 
distortion;  (2)  magnifies  without  chromatic  aberration; 
(3)  inverts  without  magnifying.  (See  references.) 

XXXVII.  MAGNIFYING  POWER  OF  A  TELESCOPE. 

Glazebrook  and  Shaw,  pp.  358—363;  Watson's  Practical  Physics,  pp. 

367,  368. 

The  magnifying  power  of  a  telescope  is  the  ratio  of  the 
angle  subtended  at  the  eye  by  the  image  as  seen  through 
the  telescope  to  the  angle  subtended  by  the  object  viewed 
directly.  (If  Exps.  XXXVI  and  XXXVIII  have  already 
been  performed,  use  the  telescope  employed  in  those 
experiments.) 

(A)  Direct  Method. — A  minute  mirror  is  attached  to  the 
telescope  by  wax  so  as  to  make  an  angle  of  about  45°  with 
the  axis  and  partly  cover  the  aperture  of  the  eye-piece. 
The  telescope  is  focused  upon  a  scale.  A  second  scale  is 
mounted  parallel  to  the  first  and  near  the  eye-piece,  in  such  a 


MAGNIFYING    POWER    OF   A    TELESCOPE.  143 

position  that  the  observer's  eye  sees,  side  by  side,  the  image 
of  the  scale  viewed  though  the  telescope  and  the  image  of 
the  other  scale  reflected  in  the  small  mirror.  From  the 
ratio  of  the  images  of  one  or  more  scale  divisions,  and  their 
distances,  the  angles  are  calculated,  and  from  their  ratio  the 
magnifying  power  is  deduced. 

Find  the  magnifying  power  for  at  least  six  distances, 
making  several  observations  for  each.  Also  determine 
the  angular  field  of  view  of  the  telescope  by  determining  for 
each  distance,  r,  the  total  distance  on  the  scale,  n,  visible 
in  the  telescope.  The  angular  field  of  view,  in  degrees,  will  be 


The  magnifying  power  defined  above  is  very  approximately 
equal  to  (i)  the  ratio  of  the  magnitude  of  the  image  to  the 
magnitude  of  the  object  when  the  two  are  in  the  same 
plane,  and,  for  great  distances,  is  equal  to  (2)  the  ratio  of 
the  focal  length  of  the  objective  to  the  focal  length  of  the 
eye-piece. 


.*• 

~~---K)* 
---''  f> 


FIG.  39. 

The  eye-piece  is  of  such  short  focus  that  the  angle  subtended  by 
its  image  is  practically  the  same  as  if  the  image  were  at  infinity.  For 
convenience  we  will  consider  the  virtual  image  P'  Q'  (see  Fig.  39) 
produced  by  the  eye-piece  to  be  at  the  same  distance  as  the  object 

Since  the  telescope  usually  views  objects  at  distances  great  com- 
pared with  its  own  length,  the  angle  subtended  by  the  object  viewed 
directly  is  practically  P  a  Q  =  p  a  q,  and  that  subtended  by  the  image 
is  P'  b  Qf  =  p  b  q.  The  ratio  of  these  two  angles,  which  may  be  taken 
as  the  ratio  of  the  tangents,  since  the  angles  are  small,  =  a  c  -r-  c  b  =  the 
ratio  of  the  focal  length  of  the  objective  to  the  focal  length  of  the 
eye-piece;  and  also,  since  the  length  of  the  telescope  is  short  compared 
with  the  distance  of  the  object,  this  ratio  =  P'  Q'  +  P  Q,  or  the  ratio 
of  the  magnitude  of  the  image  to  the  magnitude  of  the  object. 


144  LIGHT. 

(B)  The  first  approximate  statement  of  the  magnifying 
power  furnishes  another  method  for  determining  the  mag- 
nifying power  for  different   distances  of  the   object.     The 
telescope  is  directed  toward  a  horizontal  scale.     The  scale 
is  viewed  through  the  telescope  with  one  eye  and  is  also 
observed  with  the  other  eye  by  looking  along  the  outside 
of  the  telescope.     The  eye-piece  is  moved  in  or  out  until  the 
image  appears  at  the  same  distance  as  the  scale  as  viewed 
outside  the  telescope  with  the  other  eye,  i.  e.,  until  there  is 
no  parallax  between  the  scale  and  its  image   (no  relative 
motion  of  the  two  as  the  eye  is  moved  about).     It  may 
require  some  practice  to  secure  this.     Determine  the  number 
of   divisions   on   the   scale   which,    as   viewed   directly,    are 
covered  by  the  image  of  one  or  two  large  divisions  as  viewed 
through  the  telescope.     If  it  is  difficult  to  read  the  division 
on  the  scale  viewed  directly,  two  black  strips  may  be  moved 
along  the  scale  until  they  include  the  image  of  one  or  more 
divisions  as  seen  through  the  telescope,   and  the  distance 
between  these  strips  read  off.     Repeat  the  measurements 
of  (A). 

(C)  The  second  method  of  defining  the  magnifying  power 
of  a  telescope  is  useful  in  determining  the  magnifying  power 

'for  very  distant  objects.  Focus  the  telescope  on  some  very 
distant  object.  Without  changing  the  focus,  remove  the 
object  glass  and  substitute  for  it  a  diaphragm  with  a  rec- 
tangular opening.  The  ratio  of  the  focal  length  of  the 
objective  to  the  focal  length  of  the  eye-piece  is  the  ratio  of  a 
linear  dimension  of  the  aperture  of  the  diaphragm,  L,  to  the 
corresponding  dimension,  /,  of  the  image  of  this  aperture 
produced  by  the  eye-piece. 

Since  the  telescope  was  focused  for  parallel  rays,  the  distance,  u, 
of  the  object,  L,  from  the  eye-piece  is  numerically  very  nearly  the 
sum  of  the  focal  lengths,  P+f  (Fig.  40).  .'.the  distance  of  the 
image,  /,  formed  by  the  eye-piece,  is  determined  by 

i=_-i         i     -f+(F+f)          F 

v     (F+fVf-    /(F+/)       /(F+/)' 
Hence, 


RESOLVING    POWER    OF    OPTICAL    INSTRUMENTS.          145 

To  measure  /,  a  micrometer  microscope  (see  p.  15)  may 
be  used,  the  microscope  being  in  line  with  the  axis  of  the 
telescope  and  focused  upon  the  real  image  in  space.  L  may 


FIG.  40. 

be  measured  with  vernier  calipers,  or  the  same  micrometer 
microscope  may  be  placed  opposite  the  other  end  of  the 
telescope  and  L  measured  in  the  same  way  as  /. 

Questions. 

1.  Explain  why  the  magnifying  power  should  vary  as  you  have 
found  it  to  do  with  the  distance  of  the  object. 

2.  Which  is  preferable — to  gain  magnifying  power  by  increasing 
the  focal  length  of  the  objective,  or  by  decreasing  the  focal  length  of 
the  eye-piece?     Why? 


XXXVIII.  RESOLVING  POWER  OF  OPTICAL 
INSTRUMENTS. 

Text-book  of  Physics  (Duff),  pp.  421,  422;  Watson's  Practical  Physics, 
pp.  335—338;  Antes'  General  Physics,  pp.  483—487;  Mann, 
Advanced  Optics,  pp.  11-18;  Drude's  Optics,  pp.  235-236;  Hast- 
ings' Light,  pp.  70—72. 

The  magnification  obtained  with  an  optical  instrument 
depends  upon  the  focal  lengths  of  its  lenses,  as  has  been 
seen  in  the  case  of  the  telescope.  The  ability  to  distinguish 
details  of  the  image,  i.  e.,  the  "resolving  power,"  depends 
on  the  diameter  of  the  aperture  through  which  light  enters 
the  instrument. 

If  d  is  the  distance  between  two  details  of  an  object  at 
a  distance  D  from  an  aperture  whose  width  parallel  to  these 
details  is  a,  they  may  be  distinguished  if 

d      I 

D>a 

10 


146  LIGHT. 

where  /  is  the  wave-length  of  the  light  employed.     If  the 
aperture  is  circular,  the  equation  is 

d  I 


where  a  is  the  diameter  of  the  aperture. 

(A)  Resolving  Power  of  Telescope.  —  Metal  gauze  answers 
as   a   very   satisfactory   object   for   studying  the   resolving 
power,  as  it  gives  a  great  amount  of  uniform  detail.     Since 
this  detail  consists  of  rectangular  lines,  and  the  aperture  of 
the  object  glass  is  circular,  the  determination  of  the  maxi- 
mum distance  at  which  the  lines  are  discernible  will  be  more 
definite,  if  the  aperture  is  made  rectangular  by  placing  a  slit 
in  front  of  the  object  glass. 

Determine  carefully  by  several  settings,  the  maximum 
distance,  D,  at  which  the  lines  of  the  gauze,  parallel  to  the 
slit,  are  perceived.  The  gauze  should  be  illuminated  from 
behind  by  monochromatic  light  (p.  124).  Measure  carefully 
the  distance,  d,  between  the  centers  of  the  adjacent  wires 
of  the  gauze,  and  the  width  of  the  slit,  a.  Compare  d/D 
with  //a;  /  may  be  obtained  from  Table  XVIII.  Repeat 
with  other  slits  and  other  gauzes. 

(B)  Resolving  Power  of  Eye.  —  With   Porter's   apparatus 
the   resolving   power   of   the   eye   may   be   determined   for 
various  apertures. 

Four  different  gauzes,  37  .6,  27,  20,  and  14,  meshes  to  the 
cm.,  respectively,  may  be  viewed  through  four  different 
apertures  of  diameters  i.oo  mm.,  0.65  mm.,  0.53  mm., 
and  0.35  mm.,  respectively.  The  resolving  power  is  de- 
temined  by  finding  the  distance,  D,  from  a  slit  of  diameter 
a  to  a  gauze,  of  which  the  distance  between  the  centers  of 
two  adjacent  wires  is  d,  when  the  wires  are  separately  dis- 
cernible. From  the  mean  position  of  several  settings  of  a 
particular  gauze  for  a  particular  aperture,  d/D  should  be 
calculated  and  compared  with  1.2  I  /a.  If  ordinary  light  is 
used,  /  may  be  taken  as  0.00006  cm.  Use  each  aperture 
and  gauze  in  succession. 


WAVE-LENGTH    OF    LIGHT    BY    DIFFRACTION    GRATING.       147 

Questions. 

(1)  Upon  what  does  the  illumination  of  the  image  of  an  optical 
rnstrument  depend? 

(2)  When  the  diameter  of  the  pupil  of  the  eye  is  4  mm.,  how  far 
away  may  two  points  be  distinguished  which  are  o.  2  mm.  apart? 


XXXIX.  WAVE-LENGTH  OF    LIGHT  BY  DIFFRACTION 

GRATING. 

Text-book  of  Physics  (Duff),  pp.  423,  437;  Watson's  Physics,  pp. 
529-532;  Ames'  General  Physics,  pp.  530-537;  Crew's  General 
Physics,  pp.  488—491;  Edser,  Light,  pp.  448-458;  Wood,  Physical 
Optics,  pp.  168-180. 

A  diffraction  grating  consists  of  a  great  many  lines  ruled 
parallel  and  equidistant  on  a  plane  (or  concave)  surface. 
If  the  surface  be  that  of  glass,  the  grating  is  a  transmission 
grating;  if  of  metal,  a  reflection  grating.  If  a  transmission 
grating  be  placed  perpendicular  to  homogeneous  parallel 
light  from  a  collimator  (see  Exp.  XXXIII)  and  with 
the  lines  parallel  to  the  slit,  a  series  of  spectra  will  be  formed 
on  either  side  of  the  beam  of  light  transmitted  without 
deviation.  If  n  be  the  number  or  order  of  a  particular 
spectrum  counting  from  the  center,  6  the  deviation  or  angle 
that  the  rays  forming  the  spectrum  make  with  the  original 
direction  of  the  light,  a  the  grating  space  or  average  distance 
between  the  centers  of  adjacent  lines,  and  /  the  wave-length 
of  the  light 

I— —a  sin  6. 
n 

The  deviation  6  may  be  observed  by  placing  the  grating 
on  a  spectrometer  (see  Exp.  XXXIII  where  the  ad- 
justments of  the  spectrometer  are  described).  The 
position  of  the  telescope  when  in  line  with  the  collimator  is 
read.  The  grating  is  adjusted  parallel  to  the  axis  of  rotation 
of  telescope  and  collimator  as  one  face  of  a  prism  is  adjusted. 
For  convenience  in  adjusting,  the  plane  of  the  grating  should 
be  perpendicular  to  the  line  of  two  of  the  leveling-screws. 
This  enables  us  to  adjust  the  lines  of  the  grating  parallel 


148 


LIGHT. 


to  the  slit  by  means  of  one  leveling-screw  without  altering 
the  plane  of  the  grating.  The  lines  are  parallel  to  the  slit 
when  the  spectrum  of  some  homogeneous  light,  e.  g.,  from  a 
sodium  flame  (p.  124),  is  as  distinct  as  possible.  When  the 
plane  of  the  grating  is  perpendicular  to  the  incident  light,  the 
deviations  (on  opposite  sides)  of  the  two  spectra  of  the  same 
order  should  be  equal.  This  adjustment  is  also  secured 
when  that  part  of  the  beam  which  is 
reflected  back  to  the  collimator  appears 
co-axial  with  its  object  glass. 

Determine  first  the  wave-length  of 
sodium  light.  For  a  final  measure- 
ment of  the  deviation  of  any  spectrum 
the  mean  of  at  least  three  measure- 
ments on  each  side  should  be  taken. 
The  deviations  of  all  the  spectra  clearly 
visible  should  be  obtained. 

If  the  grating  space  be  not  too  small 
it  may  be  obtained  by  measurements 
on  a  dividing  engine  (p.  17),  or  with  a 
micrometer  microscope  (p.  15).  In  determining  the  grating 
space  with  the  dividing  engine,  secure  the  best  possible 
illumination  of  the  lines.  Set  the  cross-hair  of  the  micro- 
scope on  a  line  and  read  the  position  of  the  divided  head 
(circular  scale).  Watching  the  lines  through  the  micro- 
scope, turn  the  screw,  always  in  the  same  direction,  until, 
for  example,  the  tenth  line  is  under  the  cross-hair,  and  read 
the  circular  scale.  Then  turn  the  screw  until  the  tenth  line 
from  this  is  under  the  cross-hair,  read  the  scale,  and  so  on. 
Take  ten  such  groups  in  different  parts  of  the  grating. 
Find  the  average  grating  space  from  the  mean.  When  the 
grating  space  is  very  small,  the  wave-length  of  some  well- 
known  spectrum  (e.  g.,  sodium)  is  assumed  in  order  that  the 
grating  space  may  be  derived  by  reversing  the  process  of 
finding  the  wave-length. 

If  time  permit,   determine  the  wave-length  of  as  many 
other  lights  (colors)  as  possible  (see  p.  124). 


FIG.  41. 


INTERFEROMETER.  149 

Questions. 

1.  What  do   you    observe    as    regards   the    width   of   spectra    of 
different  orders?     What  would  this  indicate  as  regards  the  disper- 
sion if  mixed  light  or  light  from  an  incandescent  solid  were  used  ? 

2.  What  is  a  normal    spectrum   and  wherein   does   a   prismatic 
spectrum  differ  from  a  diffraction  spectrum?     (See  references.) 


XL.  INTERFEROMETER. 

Text-book  of  Physics  (Duff),  p.  438;  Watson's  Physics,  p.  540;  Mann, 
Advanced  Optics,  Chap.  V;  Wood,  Optics,  Chap.  VIII;  Michelson, 
Light  Waves  and  Their  Uses. 

The  interferometer  is  an  instrument  for  determining  the 
number  of  wave-lengths  of  a  monochromatic  light  con- 
tained in  a  given  distance.  For  a  description  of  the  inter- 
ferometer and  the  adjustments,  see  the  references. 

The  interferometer  will  be  used  to  determine  the  wave- 
length of  sodium  light,  assuming  a  knowledge  of  the  true 

pitch  of  the  screw.     This  will  | i 

illustrate  the  more  practical 
and  common,  but  also  more 
difficult,  utilization  of  the  in- 
terferometer in  determining  a 
length,  assuming  a  knowledge 
of  the  wave-length  of  the  light 
employed. 

The    light    5    (see    figure) 
had  best   be  monochromatic,  FlG  42 

e.  g.,  a  sodium  flame.  Initially, 

place  the  mirror  D  at  approximately  the  same  distance 
from  the  rear  face  of  A  as  the  distance  from  this  surface 
to  C.  Adjust  C  until  its  image  coincides  with  either  of 
the  images  from  D.  (There  will  be  two  images  owing  to 
reflection  from  the  two  faces  of  A.)  A  slight  adjustment 
will  now  give  the  fringes  (alternate  light  and  dark  bands, 
preferably  arcs  of  circles) .  The  observer  must  look  at  A  in 
a  direction  parallel  to  AD. 

Move  the  mirror  D  by  means  of  the  worm,   and  count 


150  LIGHT. 

the  number  of  fringes  which  pass  over  the  field  of  view. 
A  needle  in  front  of  A  may  help  as  an  index. 

From  the  number  of  turns  and  fractional  turns  of  the 
screw  and  the  value  of  the  pitch  of  the  screw,  find  the 
distance  D  has  moved  and  from  this  and  the  number  of 
fringes  which  have  passed,  calculate  the  wave-length. 
Notice  that  the  length  of  the  path  of  the  light  changes  by 
twice  the  displacement  of  the  mirror  D. 


XLI.  ROTATION    OF    PLANE    OF    POLARIZATION. 

Text-book  of  Physics  (Duff),  pp.  476-479;  Watson's  Physics,  pp.  580- 
582;  Ames'  General  Physics,  pp.  563-565;  Watson's  Practical 
Physics,  pp.  370—377;  Edser,  Light,  pp.  503—509;  Wood,  Optics, 
Chap.  XIV;  Ewell,  Physical  Chemistry,  pp.  217-223. 

Plane  polarized  light  is  obtained  by  passing  light  through 
a  Nicol  prism.  If  the  light  be  then  allowed  to  fall  on  a 
second  Nicol  prism  that  can  be  rotated,  there  will  be  two 
positions  of  this  second  prism  in  a  complete  rotation  in 
which  no  light  will  pass  through.  If  an  optically  active 
substance,  such  as  a  solution  of  cane  sugar,  be  then  intro- 
duced between  the  two  Nicols,  it  will  rotate  the  plane  of 
polarization  of  the  light  which  falls  on  the  second  prism, 
and  then,  to  quench  the  light,  the  second  prism  must  be 
rotated  through  an  equal  angle.  Thus  the  rotation  pro- 
duced by  the  sugar  is  measured. 

Monochromatic  light  must  be  used  and  a  sodium  flame 
is  most  convenient  (see  p.  124).  The  light  rays  must  be 
made  parallel  before  they  fall  on  the  polarizing  prism,  other- 
wise rays  in  different  directions  would  pass  through  different 
thicknesses  of  the  sugar  and  would  consequently  be  rotated 
by  different  amounts.  Parallel  light  may  be  obtained  by 
putting  the  source  at  the  principal  focus  of  a  convex  lens 
through  which  the  light  has  to  pass  before  falling  on  the 
polarizing  Nicol.  The  light  must  also  be  parallel  to  the  axis 
of  the  Nicol. 


ROTATION    OF    PLANE    OF   POLARIZATION.  151 

The  empty  tube  intended  to  contain  the  sugar  solution  is 
first  placed  in  position  between  the  prisms  and  the  position 
of  the  analyzing  Nicol  noted,  on  the  circular  scale,  when  the 
light  is  quenched.  This  setting  will  be  facilitated  by  using 
a  screen  to  cover  all  but  a  small  central  part  of  the  prism. 
It  may  be  found  that  the  Nicol  can  be  rotated  through  an 
appreciable  angle  without  the  light  reappearing.  The  best 
that  can  be  done  is  to  take  the  middle  of  this  space  as  the 
position  of  extinction.  The  observation  should  be  repeated 
a  number  of  times  and  the  mean  taken.  The  analyzing 
Nicol  should  then  be  rotated  through  180°  and  the  zero 
reading  in  that  position  also  noted.  Sugar  solutions  of 
different  strengths  (which  should  be  carefully  made  up  and 
recorded)  are  then  introduced  in  succession  into  the  tube  and 
the  rotations  they  produce  observed.  The  zero  readings 
should  be  frequently  repeated.  The  length  of  the  tube 
should  also  be  obtained,  so  that  the  rotation  per  decimeter 
may  be  deduced.  With  the  results  obtained,  a  curve 
should  be  plotted,  rotations  per  decimeter  being  ordinates 
and  concentrations  abscissae. 


FIG 


•  43- 


The  apparatus  here  described  is  simple  but  very  imperfect 
in  its  action.  The  sensitiveness  is  greatly  increased  by 
introducing  between  the  polarizer  and  the  specimen  a  so- 
called  biquartz,  two  parallel,  abutting,  plates  of  quartz, 
one  with  left  rotatory  power  and  the  other  with  right.  A 
source  of  white  light  must  be  employed,  for  example,  a 
frosted  incandescent  bulb,  and  the  analyzer  is  set  for  equality 
of  color  in  the  two  halves.  There  are  two  common  colors, 
but  the  darker  is  preferable. 

Fig.  44  explains  the  color  changes.  R  and  L  are  the  two  halves 
of  the  bicjuartz,  viewed  from  the  analyzer.  The  two  halves  are  of 
such  a  thickness  (3.75  mm.)  that  the  plane  of  polarization  of  yellow 


152 


LIGHT. 


light  is  rotated  through  90°.  Owing  to  the  rotatory  dispersion 
the  other  colors  will  be  rotated  different  amounts  as  shown  by  the 
letters  R  (red)  and  B  (blue).  If  the  analyzer  is  set  to  transmit  light 
vibrations  parallel  to  those  which  left  the  polarizer,  the  yellow  light 
will  be  omitted  and  each  half  of  the  biquartz  will  appear  of  a  purplish 
color  ("tint  of  passage").  If  the  analyzer 
is  displaced  slightly  clockwise,  more  of  the 
red  component  on  the  right  will  be  trans- 
mitted and  less  of  the  blue,  and  therefore 
this  half  will  appear  red  and  the  other 
half  will  appear  blue. 

If  a  dextrorotatory  specimen  is  placed 
between  the  biquartz  and  the  analyzer, 
the  directions  of  vibration  of  the  different 
colors  will  be  rotated  to  the  positions 
indicated  by  the  dotted  lines  and  the 
analyzer  must  be  rotated  to  a  new  position 
(/') ,  perpendicular  to  the  emerging  yellow 
vibration,  in  order  to  have  the  two  halves 
the  same  color.  With  the  help  of  the 
biquartz  the  analyzer  can  be  set  within 
about  a  tenth  of  a  degree. 

The     effective    thickness    of    the 
biquartz    will    not    be    correct    (90 
degrees   rotation   of   sodium   light), 
FIG.  44.  unless  it  is  perpendicular  to  the  axis 

of  the  Nicol  prisms.     This  may  be 

secured  by  using  sodium  light  and  analyzer  set  for  extinc- 
tion, and  then  placing  the  biquartz  in  such  a  position  that 
there  is  still  extinction  when  the  analyzer  is  rotated  90°. 

Repeat  all  the  measurements  with  white  light  and  the 
biquartz  and  plot  the  results  on  the  same  sheet  with  the 
preceding. 

QUESTIONS. 

1.  How  can  the  rotation  be  partially  explained  ?     (See  references.) 

2.  What   is  the   chemical  characteristic   of  substances   that   are 
optically  active  in  solution  ? 

3.  Wherein  does  the  rotation  produced  by  a  solution  differ  from 
that  produced  by  a  magnetic  field? 

4.  What  would  be  the  effect  of  using  white  light  in  the  first  part 
of  the  experiment? 


ELECTRICITY  AND  MAGNETISM. 

29.  Resistance -boxes. 

A  resistance-box  consists  of  a  number  of  resistance  coils 
joined  so  that  each  one  bridges  the  gap  between  two  of  a 
series  of  brass  blocks  placed  in  line  on  the  cover  of  the  box 
within  which  the  coils  are  suspended.  For  each  gap  a  plug 
or  connector  is  also  provided,  and  when  the  plug  is  inserted 
into  the  gap  the  resistance  at  the  gap  is  "  cut  out "  or  practi- 
cally reduced  to  zero.  The  coils  are  wound  so  as  to  be  free 
from  self-induction.  The  successive  resistances  are  arranged 
in  the  same  order,  and  are  of  the  same  relative  magnitudes 
as  the  successive  weights  in  a  box  of  weights.  By  removing 
the  proper  plugs  any  combination  of  resistances  can  be 
obtained  from  the  smallest  to  the  sum  of  all.  Before  begin- 
ning work,  it  is  advisable  to  clean  the  plugs  with  fine  emery- 
cloth  so  that  they  may  make  good  contacts,  and  thereafter 
care  should  be  taken  not  to  soil  them  with  the  fingers. 

One  important  precaution  in  regard  to  the  use  of  the 
resistance-box  should  be  observed.  If  any  of  the  plugs 
are  in  loosely,  there  will  be  some  resistance  at  the  contact. 
Hence,  the  plug  should  be  screwed  in  firmly,  but  not  vio- 
lently. When  any*  one  plug  has  been  withdrawn,  the  others 
should  be  tested  before  proceeding,  for  the  removal  of  one 
may  loosen  the  contact  of  the  others.  This  precaution  is 
especially  important  in  making  a  final  determination. 

30.  Forms  of  Wheatstone's  Bridge. 

The  practical  measurement  of  a  resistance  consists  in 
comparing  it  with  a  known  or  standard  resistance.  Wheat- 
stone's  Bridge  is  an  arrangement  of  conductors  for  facilitat- 
ing this  comparison,  and  consists  essentially  of  six  branches 

153 


ELECTRICITY  AND    MAGNETISM. 


which  may  be  represented  by  the  sides  and  diagonals  of  a 
parallelogram  (see  Fig.  45).  The  unknown  resistance, 
R,  and  the  known  resistance,  5,  form  two  adjacent  sides. 
The  other  two  sides  are  formed  by  two  conductors  of  resist- 
ance P  and  Q,  which,  however,  do  not  need  to  be  known 
separately,  provided  their  ratio  be  known.  One  of  the- 

diagonals  contains  a  battery 
and  the  other  a  galvanom- 
eter. If  the  ratio  of  P  to 
Q  is  adjusted  until  no  current 
flows  through  the  galvanom- 
eter, R:S::P:Q.  (See  refer- 
ences under  Exp.  XLIV.) 

Two  forms  of  the  Wheat- 
stone  "  Bridge  arrangement 
are  in  common  use.  One  is 
called  the  Wire  (or  meter) 
Bridge;  the  other,  which  uses 


/r 


FIG.  45. 
is  called  a  Bridge  Box. 


a  box  of  adjusted  resistances, 
In  the  wire  bridge  the  "ratio 
arms"  (whose  resistances  are  P  and  Q)  are  the  two  parts  of 
a  uniform  wire  i  meter  long,  and  the  ratio  of  P  to  Q  is  that 
of  the  lengths  of  the  corresponding  parts  of  the  wire.  The 
known  resistance,  5,  may  be  that  of  a  standard  coil  or  one 
of  the  known  resistances  of  a  resistance-box. 

The  Bridge  Box,  or  "Post-office  Bridge,"  consists  of  a 
resistance-box  with  three  series  of  resistances  in  line,  forming 
three  arms  of  the  Wheatstone  Bridge,  the  unknown  resist- 
ance forming  the  fourth  arm.  The  "ratio  arms"  consist  of 
resistances  of  i,  10,  100,  1000  (all  of  which  are  not  always 
necessary),  so  that  the  calculation  of  the  ratio  is  very 
simple.  Keys  for  closing  the  battery  and  galvanometer 
branches  are  also  usually  mounted  on  the  box. 


GALVANOMETERS.  155 

31.  Galvanometers. 

Text-book  of  Physics  (Duff),  pp.  559-563;  Hadley's  Electricity  and 
Magnetism,  pp.  273-284;  Watson's  Practical  Physics,  §§170-174; 
Ames  and  Bliss,  Appendix  iii. 

There  are  two  chief  types  of  reflecting  galvanometers. 
In  both  the  principle  at  basis  is  that  if  a  magnet  be  placed 
in  the  plane  of  a  coil  of  insulated  wire,  on  passing  a  current 
through  the  coil  both  magnet  and  coil  become  subject  to 
forces  that  tend  to  set  them  at  right  angles  to  each  other. 
In  the  Thomson  type  the  coil  is  fixed  and  the  magnet 
suspended  within  the  coil  is  free  to  turn,  while  in  the 
d'Arsonval  type  the  magnet,  of  a  horseshoe  form,  is  fixed, 
and  the  coil,  suspended  between  the  poles  of  the  magnet, 
is  free  to  turn. 

The  sensitiveness  of  the  Thomson  galvanometer  is  greatly 
increased  in  two  ways:  first,  two  magnetic  needles,  forming 
an  astatic  pair,  are  attached  to  the  same  axis  of  rotation, 
second,  an  external  control  magnet  is  used  to  weaken  the 
restraint  of  the  earth's  magnetic  force  or  even  to  overcome" 
the  earth's  field  and  produce  a  suitable  field  of  its  own. 
The  chief  difficulty  in  greatly  increasing  the  sensitiveness 
by  means  of  the  control  magnet  is  that  slight  variations  of 
the  whole  magnetic  field,  due  to  outside  currents  or  move- 
ments of  magnetic  materials  in  or  near  the  laboratory, 
disturb  the  needle. 

On  the  d'Arsonval  galvanometer  variations  of  the  external 
magnetic  field  have  practically  no  effect,  since  its  own 
magnetic  field  is  very  strong.  On  the  other  hand,  the 
torsion  of  the  fine  suspending  wire  through  which  the 
current  has  to  pass  changes  somewhat  with  the  temperature, 
so  that  the  zero  reading  of  the  galvanometer  is  subject  to 
some  change.  The  sensitiveness  can  be  increased  by  in- 
creasing the  strength  of  the  magnet,  but  there  is  a  limit  to 
this,  since  small  traces  of  iron  are  always  present  in  the 
wire  and  insulation  of  the  coil,  and  this,  acted  on  by  the 
magnetic  field,  exercises  a  magnetic  control  that  is  pro- 
portional to  the  square  of  the  strength  of  the  field.  When 


156  ELECTRICITY   AND    MAGNETISM. 

an  extremely  sensitive  galvanometer  for  very  accurate 
work  is  required,  the  Thomson  type  must  be  used. 

A  ballistic  galvanometer  is  a  reflection  galvanometer  of 
either  type,  so  made  that  its  period  of  swing  is  very  long, 
so  that  it  starts  into  motion  only  very  slowly.  If  this  con- 
dition be  fulfilled,  and  if  it  be  subject  to  only  very  slight 
damping  of  its  motion,  the  galvanometer  may  be  used  for 
comparing  quantities  of  electricity  suddenly  discharged 
through  the  coils  of  the  galvanometer,  for,  practically 
speaking,  all  the  electricity  will  have  passed  before  the 
swinging  system  has  appreciably  moved  from  this  position 
of  rest.  In  these  circumstances  it  can  be  shown  that  the 
quantity  of  electricity  is  proportional  to  the  sine  of  half  the 
angle  of  the  first  swing,  or  (since  the  angle  is  very  small) 
practically  to  the  deflection  as  read  on  the  scale. 

Two  methods  of  reading  the  deflection  of  a  galvanometer 
are  in  common  use.  In  one,  called  the  English  or  objective 
method,  a  beam  of  light  reflected  from  the  mirror  of  the 
galvanometer  falls  on  a  scale,  forming  a  spot  of  light  which 
moves  as  the  needle  or  coil  is  deflected.  In  the  other, 
called  the  German  or  subjective  method,  the  image  of  a  scale 
formed  by  the  mirror  of  the  galvanometer  is  read  by  a 
telescope  with  a  cross-hair. 

Devices  for  Bringing  a  Galvanometer  to  Rest. — For  bringing 
to  rest  the  needle  of  a  ballistic  Thomson  galvanometer  a 
coil  is  mounted  on  the  outside  of  the  galvanometer  in  front 
of  the  lower  needle.  The  terminals  of  the  coil  are  brought 
to  a  reversing  switch  by  which  the  current  from  a  cell  can 
be  sent  through  the  coil  in  either  direction.  By  suitably 
choosing  the  direction  and  duration  of  the  current,  the 
needles  and  mirror  may  be  brought  to  rest.  (A  current 
in  this  coil  affects  the  needle  in  the  same  manner  as  would  a 
current  in  one  of  the  regular  galvanometer  coils,  but  it  is 
much  more  convenient  to  use  a  separate  coil  like  this,  which 
is  readily  accessible  and  which  does  not  interfere  with  the 
other  connections.) 

The   suspended   system   of  either  type   of   galvanometer 


GALVANOMETER    SHUNTS.  157 

may  also  be  brought  to  rest  by  short-circuiting  the  galvan- 
ometer by  a  simple  key  directly  connected  to  the  terminals. 
For,  by  Lenz's  Law,  the  currents  induced  are  such  as  to 
bring  the  moving  coil  or  needle  to  rest.  If  the  resistance 
of  the  coils  is  high,  this  method  is  slow,  and  the  following 
more  rapid  method  may  be  used.  A  coil  in  which  a  small 
bar  magnet  can  be  moved  is  placed  in  series  with  the  short- 
circuiting  key.  By  suitably  moving  the  magnet  in  and  out, 
currents  are  induced  which  will  quickly  bring  the  suspended 
system  to  rest. 

32.  Correction   for   Damping   of   a    Ballistic    Galvanometer- 

Kohlrausch's  Physical  Measurements,  §51;  Stewart  and  Gee's  Practical 
Physics,  II,  pp.  364—369. 

In  considering  the  throw  proportional  to  the  charge 
passing  through  the  coils  of  a  ballistic  galvanometer,  we 
assume  that  the  galvanometer  is  free  from  damping;  i.  e., 
that  the  suspended  system,  needles,  mirror,  etc.,  experiences 
no  resistance  to  turning.  Since  this  is  never  realized,  a 
correction  must  be  applied  to  the  throw. 

The  correction  is  not  of  importance  where  we  compare 
throws,  since  the  correction  cancels  out,  but  in  much 
work  with  ballistic  galvanometers  this  correction  is  very 
important. 

Set  the  needle  vibrating  and  record  n  +  i  successive  turn- 
ing-points. From  these  we  obtain  by  successive  subtrac- 
tion n  successive  full  vibrations  of  the  needle  from  one 
side  to  the  other.  Call  the  first  full  vibration  al  and  the 
last  an.  Then  the  correction  by  which  each  throw  should  be 
multiplied  is  (i  -M/2)  where 

^^loga1-logqM 

n  —  i 

\ 

33.  Galvanometer  Shunts. 

If  a  galvanometer  of  resistance  G  is  shunted  by  a  shunt 
of  resistance  5  and  if  C  is  the  whole  current  and  Cl  the 
current  through  the  galvanometer 


ELECTRICITY   AND    MAGNETISM. 


C     G+S' 

Galvanometers  are  frequently  supplied  with  shunt-boxes 
in  which  the  ratio  of  5:  G  are  1/9,  1/99,  1/999,  so  "that  the 
values  of  5  :(G  +S)  are  i/io,  i/ioo, 
i/iooo.  Such  a  shunt-box  cannot 
easily  be  used  with  any  galvanom- 
eter except  that  for  which  it  was 
designed. 

Universal  shunt-boxes  are  now 
made  which  can  be  used  with  any 
galvanometer.  Such  a  box  consists 
of  a  series  of  high,  resistances  con- 
nected as  indicated  in  the  figure.  AB 
is  a  coil,  of  resistance  S,  connected  to  the  galvanometer,  of 
resistance  G.  Let  the  current  through  the  galvanometer 
be  Clt  and  let  the  whole  current  be  C.  Then  as  above 

c,=  cs 


VVvNA/VV\AAAA/V  j? 


FIG.  46. 


Now  let  the  battery  circuit  be  connected  to  A  and  P,  where 
the  resistance  of  AP  is  S/n.  Denoting  the  current  through 
the  galvanometer  by  C/,  and  the  whole  current  by  C  and 
making  the  proper  changes  in  the  above  equation, 

Cy_  Sin  _i      5 

~C  ~S/n  +  (S-S/n)+G ~n  S+G' 
i    CS 


nS+G 

Hence  when  a  current  is  connected  to  A  and  P,  the  galvan- 
ometer deflection  is  i/n  as  great  as  when  the  same  current 
is  connected  to  A  and  B,  or  the  sensitiveness  is  i/n  as  great. 
By  subdividing  AB,  the  values  of  3,  10,  100,  etc.,  are  given 
to  n. 

Shunting  a  Ballistic  Galvanometer. — The  formulas  stated 
above  were  deduced  from  Ohm's  Law  for  steady  direct 
currents.  It  can,  however,  be  shown  that  shunts  like  the 


STANDARD    CELLS.  159 

above  may  be  used  in  the  same  way  with  ballistic  galvan- 
ometers through  which  charges  of  electricity  are  passed. 
To  prove  this,  all  we  need  to  do  is  to  show  that  charges, 
like  steady  direct  currents,  divide  in  a  parallel  arc  into  parts 
inversely  as  the  ohmic  resistances.  Consider  any  one  of 
several  branches  in  a  parallel  arc.  Let  the  part  of  the  charge 
that  passes  through  it  be  q,  and  let  the  magnitude  of  the 
instantaneous  current  through  it,  at  time  t  after  the  begin- 
ning of  the  discharge,  be  *\.  The  induced  e.  m.  f.  at  that 
moment  is  L^dijdt  where  Lt  is  the  self -inductance  of  the 
branch.  Suppose  the  discharge  is  caused  by  connecting  an 
e.  m.  f.  £  to  the  parallel  arc  for  a  short  time  and  then  dis- 
connecting it,  and  let  the  whole  time  of  rise  and  fall  of  the 
brief  current  be  T.  Then 


E  CT        L    C° 

-!  dt-±\  div 

Rijo  Rijo 


ET 


Hence  the  charges  through  the  various  branches  are  in- 
versely as  their  ohmic  resistances.  If  the  above  proof  be 
carefully  examined,  it  will  be  seen  that  it  simply  means 
that  the  total  quantity  due  to  the  induced  e.  m.  f  .  is  zero,  since 
the  induced  current  in  the  first  half  of  the  process  is  op- 
posite to  that  in  the  second  half. 

34.  Standard  Cells. 

Text-book  of  Physics  (Duff),  pp.  584-585;  Watson's  Physics,  pp.  806- 
807;  Watson's  Practical  Physics,  §§  202-203;  Bureau  of 
Standards  Bulletin,  Nos.  67,  70,  71;  Henderson's  Electricity  and 
Magnetism,  pp.  176-182.  Ewell,  Physical  Chemistry,  pp.  334-336. 

The   standard  Daniell  cell  consists   of  an   amalgamated 
zinc  rod  dipping  into  a  porous  cup  containing  a  solution  of 


l6o  ELECTRICITY   AND    MAGNETISM. 

sulphate  of  zinc,  which,  in  turn,  stands  in  a  glass  vessel 
containing  a  copper  sulphate  solution  and  a  copper  plate. 
To  amalgamate  the  zinc  rod,  thoroughly  clean  it  with  sand- 
paper, dip  it  in  dilute  sulphuric  acid,  and  rub  over  it  a  few 
drops  of  mercury  with  a  cloth.  The  porous  cup  should  be 
thoroughly  cleaned  inside  and  out.  The  copper  plate  should 
be  cleaned  bright  with  sand-paper.  The  porous  cup  is  half- 
filled  from  a  stock  bottle  with  a  solution  of  zinc  sulphate 
(44.  7  g.  of  crystals  of  c.  p.  zinc  sulphate  dissolved  in  100  c.c. 
of  distilled  water).  The  zinc  rod  is  introduced  and  the 
porous  cup  is  placed  in  the  glass  vessel,  which  is  filled, 
not  quite  up  to  the  level  of  the  zinc  sulphate  in  the  porous 
cup,  with  copper  sulphate  solution  (39.4  g.  of  c.  p.  copper 
sulphate  dissolved  in  100  c.c.  of  distilled  water).  The 
copper  plate  is  also  placed  in  the  outer  vessel.  After  being 
set  up,  the  cell  should  be  short-circuited  for  15  minutes 
and  then  allowed  to  stand  on  an  open  circuit  for  5  minutes. 
The  cell  should  not  remain  set  up  more  than  a  few  hours. 
When  it  is  no  longer  needed,  pour  the  copper  sulphate  solu- 
tion back  into  the  stock  bottle  and  the  zinc  sulphate  solution 
back  into  its  bottle,  unless  the  zinc  has  turned  black,  in  which 
case  throw  the  zinc  sulphate  away.  The  e.  m.  f.  of  the 
Daniell  cell,  prepared  as  above,  is  i .  105  international  volts, 
correct  to  o .  2  per  cent. 

The  Clark  cell,  which  differs  from  the  above  in  the  fact 
that  the  copper  is  replaced  by  mercury  and  the  copper  sul- 
phate by  mercurous  sulphate,  is  a  more  constant  standard 
than  the  Daniell  cell,  but  it  needs  to  be  treated  with  much 
greater  care,  since  the  passage  of  a  very  small  current  through 
it  will  alter  the  e.  m.  f.  Hence  it  can  be  used  only  for  null 
methods  and  kept  in  circuit  for  the  briefest  time  possible. 
At  temperature  /  its  e.  m.  f.  in  volts  is 

i.433-.ooi2(/-i5). 

In  the  cadmium  cell  the  zinc  and  zinc  sulphate  of  the 
above  are  replaced  by  cadmium  and  cadmium  sulphate. 
Its  e.  m.  f.  is 

1.019  —  .  00004(2  — 17). 


DOUBLE    COMMUTATOR. 


35.  Device  for  Getting  a  Small  E.  M.  F. 

In  many  experiments  it  is  desirable  to  use  an  e.  m.  f- 
much  smaller  than  that  of  a  single  cell.  To  get  such  an 
e.  m.  f.,  a  box  of  very  high  resistance  may  be  placed  in 
series  with  a  constant  cell  and  any  desired  fraction  of  the 
whole  e.  m.  f.  may  be  obtained  by  tapping  off  from  various 


FIG.  47- 

points;  e.  g.,  at  the  ends  of  a  resistance  r  out  of  the  total  re- 
sistance R  of  the  box  (Fig.  47).  The  e.  m.  f.  thus  obtained 
may  be  found  from  Ohm's  Law,  but  it  must  be  noticed 
that  the  resistance  between  the  terminals  of  r  is  the  resist- 
ance of  a  parallel  arc.  If,  however,  the  resistance  of  the 
branch  circuit  be  proportionally  very  large  and  that  of  the 
cell  proportionally  very  small,  both  may  be  omitted  in  the 
calculation. 

36.  Double  Commutator. 

It  is  sometimes  desirable  to  be  able  to  reverse  two  parts 
of  a  network  repeatedly  and  at  the  same  rate.  For  this  pur- 
pose a  double  commutator  is  con- 
venient. It  consists  of  two  two-part 
commutators  mounted  on  a  common 
shaft;  e.  g.,  on  opposite  ends  of  the 
shaft  of  a  small  motor.  If,  for  ex- 
ample, the  battery  used  with  a  Wheat- 
stone's  Bridge  be  connected  through 


one  commutator  while  the  galvanom 
eter   is  connected  through  the  other, 
ii 


FIG.  48. 


1 62  ELECTRICITY   AND    MAGNETISM. 

an  alternating  current  will  act  in  the  arms  of  the  bridge, 
while  a  direct  current  (or  a  succession  of  unidirectional 
pulses)  will  pass  through  the  galvanometer. 

37.  Relation  Between  Electrical  Units. 

(E.S.  =  Electrostatic;  E.M.  =  Electromagnetic.) 
Ampere       =io-1  C.G.S.-E.M.  units  of  current. 
Coulomb     =io-1  C.G.S.-E.M.  units  of  quantity. 
Volt  =io8     C.G.S.-E.M.  units  of  electromotive  force. 

Ohm  =  io9     C.G.S.-E.M.  units  of  resistance. 

Farad  =io-9  C.G.S.-E.M.  units  of  capacity. 

Microfarad  =  io-15C.G.S.-E.M.  units  of  capacity. 
Henry          =io9     C.G.S.-E.M.  units  of  inductance. 
Volt  =£xio-2  C.G.S.-E.S.  units  of  electromotive  force. 

Coulomb     =3X10"     C.G.S.-E.S.  units  of  quantity. 
Microfarad  =  9  Xio5     C.G.S.-E.S.  units  of  capacity. 


XLII.  HORIZONTAL    COMPONENT    OF    EARTH'S 
MAGNETIC  FIELD. 

Ames1  General  Physics,  pp.  609—613;  Watson's  Physics,  pp.  602-607; 
Text-book  of  Physics  (Duff),  pp.  498-500;  Crew' s  General  Physics , 
pp.  319-323;  Hartley's  Electricity  and  Magnetism,  pp.  92-98; 
Watson's  Practical  Physics,  pp.  403-414;  Kohlrausch's  Physical 
Measurements,  pp.  240-247;  Stewart  and  Gee's  Practical  Physics, 
pp.  284-309- 

In  this  experiment  the  horizontal  component  of  the 
earth's  magnetic  field,  at  a  point  in  the  laboratory,  is  de- 
duced from  the  period  of  vibration  of  a  bar-magnet  and  the 
deflection  of  a  magnetic  needle  produced  by  this  same  bar- 
magnet  when  placed  at  known  distances  E  and  W  (mag- 
netically) of  the  needle.  The  dimensions  and  mass  of  the 
magnet  must  also  be  obtained  in  order  that  its  moment  of 
inertia  may  be  calculated. 

If  the  period  of  vibration  of  the  magnet  be  T  in  the 
place  in  which  we  wish  to  determine  the  horizontal  com- 
ponent H,  its  magnetic  moment  be  M,  and  its  moment  of 
inertia  /,  then 


d) 

•\M# 

For,  when  the  magnet  is  deflected  through  a  small  angle  6,  the 
restoring  couple  is  MH  sind  =  MH6.  Hence  if  the  angular  accelera- 
tion at  that  moment  is  a 

-MH6=Ia 
and 


Since  M,  H,  and  7  are  constant,  the  motion  is  simple  harmonic  and 
T  is  given  by  (i). 

If  a  magnetic  needle  at  a  distance  d,  E  or  W  of  this  same 
bar-magnet,  in  line  with  it  and  the  point  where  H  is  to  be 


164  ELECTRICITY   AND    MAGNETISM. 

determined,   be   deflected   through   an   angle   <p  and,  when 
at  a  distance  d^  be  deflected  through  an  angle  (^ 

M     d5  tan  (b—d*  tan  d>\ 

(->)  —  = L- l 

H  2(d2-d,2) 

Equation  (2)  is  deduced  from  the  expression  for  the  force  F  pro- 
duced by  a  magnet  of  magnetic  moment  M  at  a  distance  d  in  the 
direction  of  the  axis  of  the  magnet.  For,  if  m  is  the  strength  of  either 
pole  of  the  magnet  and  2 1  its  magnetic  length,  the  resultant  force 
due  to  the  two  poles  is 


By  expanding  the'denominator  we  may  also  write  this : 
F  =  *4 


in  which  K  is  approximately  a  constant.  (If  the  length  of  the  needle 
were  also  taken  account  of,  this  expression  would  remain  unchanged , 
except  that  the  value  of  the  constant  K  would  be  different) .  If,  un- 
der the  force  F  and  the  component  H  of  the  earth's  magnetic  field, 
a  magnetic  needle  makes  an  angle  0  with  the  magnetic  meridian, 
F  =  H  tan  0.  Hence, 

M  /       K\      d*  tan  d> 


If,  now,  the  distance  be  changed  to  dlt  and  the  deflection  becomes 
</>t  another  equation  similar  to  the  above  will  be  obtained  and  the 
elimination  of  K  will  give  equation  (2)  above. 

From  equations  (i)  and  (2)  both  H  and  M  may  be  obtained 
when  the  other  quantities  have  been  measured. 

(A)  To  determine  the  period  of  vibration,  remove  all 
movable  iron  (knives,  keys,  etc.,  included)  to  several 
meters  from  the  vicinity  of  the  entire  experiment.  Suspend 
the  deflecting  magnet,  by  means  of  a  stirrup  attached  to  a 
single  strand  of  silk  thread,  in  a  box  which  has  glass  ends 
and  sides  and  is  surmounted  by  a  glass  tube  through  which 
the  suspension  passes.  Level  until  the  thread  hangs  in  the 
axis  of  the  tube.  The  magnet  should  be  adjusted  until  it  is 
horizontal  as  tested  by  comparison  with  a  leveled  rod 
attached  to  the  outside  of  the  box.  Attach  pointers  to  the 
opposite  glass  sides  of  the  box  (or  adjust  those  provided) 
so  that  they  are  in  line  with  the  magnet  at  rest.  Set  the 
magnet  vibrating  through  an  angle  not  exceeding  10°. 


HORIZONTAL  COMPONENT  OF  EARTH'S  MAGNETIC  FIELD.    165 

Check  any  pendulum  vibrations  by  judiciously  pressing  on 
the  top  of  the  glass  tube.  Then  determine  the  period  by  the 
method  of  passages  as  in  Exp.  X.  (see  p.  54). 

The  magnet  should  be  vibrated  as  near  as  is  convenient 
to  the  place  where  the  needle  is  deflected  in  the  second 
part,  i.  e.,  where  we  wish  to  determine  H. 

(B)  The  instrument  used  in  the  deflection  part  of  the 
experiment  is  called  a  magnetometer.  It  consists  of  a  box 
with  glass  sides  in  which  is  suspended  a  mirror  attached  to 
either  a  small  magnetic  needle  with  a  damping  vane  or  a 
small  bell  magnet  vibrating  in  a  copper  sphere.  The  sphere 
is  placed  at  the  center  of  a  graduated  bar  upon  which  can 
be  placed  the  deflecting  magnet.  Level  until  the  suspending 
fiber  is  at  the  center  of  the  bottom  of  the  suspension-tube. 
If  the  needle  or  the  damping  vane  does  not  swing  free,  a 
little  additional  leveling  will  be  necessary. 

A  specially  mounted  large  compass  needle  is  used  to 
adjust  the  magnetometer  bar  perpendicular  to  the  magnetic 
meridian.  By  means  of  it  a  rod  is  placed  in  the  direction 
of  the  magnetic  meridian,  and  then,  by  means  of  a  square, 
the  magnetometer  bar  is  made  perpendicular  to  the  rod. 
Place  a  telescope  and  scale  about  one  meter  from  the  mag- 
netometer. See  that  the  scale  is  perpendicular  to  the 
telescope.  Adjust  until  the  scale  reflected  from  the  mirror 
is  clearly  seen  in  the  telescope  (for  directions  for  this  adjust- 
ment see  p.  25). 

Place  the  magnet  whose  period  of  vibration  has  been  de- 
termined on  a  small  wood  slide  near  one  end  of  the  magnet- 
ometer bar.  Note  the  scale-reading  on  the  magnetometer 
bar  corresponding  to  the  end  of  the  magnet  nearer  the 
needle.  When  the  needle  comes  to  rest,  record  the  scale- 
reading  against  the  vertical  cross-hair  of  the  telescope. 
Remove  the  magnet  several  meters  and  read  the  zero. 
Replace  the  magnet  at  the  same  distance  from  the  needle, 
but  reversed,  and  again  read  the  scale  division  correspond- 
ing to  the  vertical  cross-hair.  Make  two  similar  readings 
with  the  magnet  at  an  equal  distance  on  the  other  side  of 


1  66  ELECTRICITY  AND    MAGNETISM. 

the  needle.  Read  the  zero  before  or  after  each  reading  and 
always  estimate  tenths  of  millimeters.  Make  four  similar 
readings  with  the  magnet  at  about  two-thirds  the  distance 
on  each  side  of  the  needle. 

If  the  zero  is  somewhat  unsteady,  the  following  method 
will  be  found  better.  Omit  zero  readings  and  obtain  the 
four  deflection  readings  as  rapidly  as  possible.  Repeat  this 
twice  so  that  twelve  readings  in  all  are  obtained.  Take 
half  the  difference  of  each  two  successive  readings  as  one 
value  of  the  deflection.  The  final  result  will  be  the  mean 
of  all  values  so  found.  The  extent  to  which  they  agree 
will  indicate  the  reliability  of  the  mean. 

Measure  the  distance  from  the  center  of  the  scale  beneath 
the  telescope  to  the  center  of  the  suspension-tube  of  the 
magnetometer  (i.  e.,  the  distance  to  the  mirror).  From 
this  distance  and  the  mean  scale-reading  for  that  distance, 
tan  2$  is  obtained  (for  it  must  be  remembered  that  a  re- 
flected ray  of  light  is  turned  through  twice  the  angle  that 
the  reflecting  mirror  is  turned  through).  Since  </)  is  a  small 
angle  tan  2(f>  =  2  tan  (/>  very  nearly.  The  distances  from  the 
needle  to  the  near  end  of  the  magnet  plus  half  the  length 
of  the  magnet  give  d  and  d^.  At  the  close  of  the  experiment, 
measure  the  length  of  the  magnet  with  vernier  calipers, 
and  the  diameter  with  micrometer  calipers,  and  also  weigh 
it.  If  /  be  the  length,  r  the  radius,  and  m  the  mass,  the 
moment  of  inertia  is  : 


In  reporting,   state  the  possible  errors  of  the  measure- 
ments of  r,  /,  d,  dlt  tan  <£,  tan  fa. 

Questions. 

i  .  How  could  the  true  length  of  the  deflecting  magnet  be  obtained  ? 

2.  H  having  been  obtained  at  one  point  in  the  room  or  building, 
what  would  be  the  easiest  way  of  finding  its  value  at  any  other  point  ? 

3.  What  are  the  other  "elements"  of  the  earth's  magnetism? 

4.  If  you  have  done  Exp.  XLIII,  calculate   the  total   force   and 
the  vertical  component. 


MAGNETIC    INCLINATION    OR    DIP.  167 

XLIII.  MAGNETIC  INCLINATION  OR  DIP. 

Ames'  General  Physics,  pp.  618—619;  Watson's  Physics,  pp.  605-607; 
Text-book  of  Physics  (Duff),  p.  506;  Crew's  General  Physics,  p. 
312;  Hartley's  Electricity  and  Magnetism,  pp.  99—102;  Watson's 
Practical  Physics,  pp.  4 1 5-4 1 7 ;  Stewart  and  Gee's  Practical  Physics, 
II,  pp.  275-284. 

(A)  The  dip,  or  inclination  of  the  earth's  magnetic  lines 
of  force  to  the  horizontal,  is  found  by  means  of  a  dipping 
needle  or  magnetic  needle  suspended  on  a  horizontal  axis 
which  passes  as  nearly  as  possible  through  the  center  of 
gravity  of  the  needle,  with  a  vertical  graduated  circle  for 
reading  the  angle  of  inclination.  Such  an  apparatus  is 
called  a  dip-circle,  and  includes  a  level  and  leveling  screws 
for  making  the  circle  vertical,  knife-edges  for  bearing  the 
axis  of  the  needle,  a  horizontal  graduated  circle  for  fixing 
the  azimuth  of  the  vertical  circle,  and  an  arrestment,  with 
Y-shaped  supports,  for  raising  and  lowering  the  needle  and 
placing  it  so  that  its  axis  of  rotation  passes  as  nearly  as 
possible  through  the  center  of  the  vertical  circle. 

The  zero-line  of  the  vertical  circle  must  first  be  made 
vertical.  This  adjustment  is  made  by  means  of  the  level- 
ing screws  and  level  just  as  a  cathetometer  is  leveled  (see 
p.  19).  The  circle  must  then  be  turned  into  the  plane 
of  the  magnetic  meridian.  To  attain  this,  advantage  is 
taken  of  the  fact  that  if  the  plane  in  which  the  needle  is 
free  to  rotate  be  at  right  angles  to  the  magnetic  meridian, 
the  needle  must  stand  vertically;  for  in  that  position  the 
horizontal  component  of  the  earth's  magnetic  force  is  par- 
allel to  the  axis  of  rotation  of  the  needle,  and  hence  has  no 
moment  about  that  axis.  The  circle  is,  therefore,  turned 
approximately  east  and  west  and  then  adjusted  until  the 
needle  is  vertical.  This  adjustment  should  be  repeated 
several  times,  and  each  position  should  be  carefully  read 
with  the  assistance  of  a  vernier  if  one  is  provided.  A  rota- 
tion of  the  circle  through  90°  from  the  mean  position, 
as  indicated  by  the  horizontal  circle,  should  then  bring  the 
plane  of  the  circle  to  coincidence  with  the  plane  of  the 


1 68  ELECTRICITY    AND    MAGNETISM. 

magnetic  meridian.  By  raising  and  lowering  the  arrest- 
ment,  the  needle  is  then  placed  on  the  knife-edges  in  the 
proper  position  for  indicating  the  dip. 

A  single  reading  of  the  needle  in  this  position  would  give 
a  very  imperfect  value  of  the  dip.  Errors  arise  from 
various  causes:  (i)  the  axis  may  not  roll  freely  on  the 
knife-edges,  owing  to  dust  or  friction.  To  remove  any 
dust  the  axis  and  knife-edges  should  be  brushed  with  a 
camel's-hair  brush.  The  setting  by  means  of  the  arrest- 
ment  and  the  readings  should  be  made  at  least  twice,  and 
both  sets  of  readings  recorded.  (2)  The  axis  of  rotation 
of  the  needle  may  not  be  exactly  at  the  center  of  the  divided 
circle.  This  error  may  be  eliminated  by  reading  the  posi- 
tion of  both  ends  of  the  needle,  one  reading  being  from  this 
cause  as  much  too  great  as  the  other  is  too  small.  (3)  The 
line  of  zeros  on  the  vertical  scale  may  not  be  truly  vertical, 
and  this  would  cause  errors  in  the  same  direction  in  the 
readings  of  the  ends  of  the  needle.  These  errors  may  be 
eliminated  by  turning  the  vertical  circle  through  180°  about 
a  vertical  axis  and  repeating  the  readings,  for  in  these  read- 
ings the  quadrants  on  the  other  side  of  the  zero  line  are 
used.  (4)  The  axis  of  rotation  may  not  pass  exactly  through 
the  center  of  gravity  of  the  needle.  So  far  as  the  fault 
lies  in  the  fact  that  the  axis  of  rotation  is  to  one  side  of  the 
axis  of  figure  of  the  needle,  the  error  may  be  eliminated 
by  reversing  the  needle  in  its  bearings  and  repeating  the 
readings;  for  in  one  position  gravity  will  make  the  readings 
as  much  too  great  as  in  the  other  case  it  makes  them  too 
small.  But  gravity  will  also  cause  an  error  if  the  axis 
of  rotation  be  in  the  axis  of  figure,  but  not  at  the  center  of 
the  latter.  The  error  will  not  be  eliminated  by  reversing 
the  needle  on  its  bearings,  but  it  will  be  if  the  magnetism 
of  the  needle  is  reversed  and  all  of  the  preceding  readings 
repeated;  for  then  the  other  end  of  the  needle  will  be  lower 
and  the  error  will  be  in  the  opposite  direction.  The  re- 
versal of  the  magnetism  should  be  done  under  the  direction 
of  the  instructor,  the  method  of  double  touch  being  used. 


MEASUREMENT    OF    RESISTANCE.  169 

In  recording  these  various  positions  and  readings,  the 
side  of  the  circle  on  which  the  scale  is  engraved  may  be 
called  the  face  of  the  instrument,  and  similarly  one  side  of 
the  needle  may  be  fixed  upon  as  its  face.  Thus  two  readings 
of  each  end  of  the  needle  are  to  be  made  in  each  of  the  fol- 
lowing positions: 

(1)  Face  of  instrument  E,  face  of  needle  E; 

(2)  Face  of  instrument  W,  face  of  needle  W; 

(3)  Face  of  instrument  W,  face  of  needle  E; 

(4)  Face  of  instrument  E,  face  of  needle  W. 

The  magnetism  of  the  needle  having  been  reversed, 
readings  are  to  be  again  taken  in  the  above  positions.  The 
final  result  is  taken  as  the  mean  of  these  32  readings. 

(B)  Another  method  of  determining  the  dip  is  by  means 
of  an  earth  inductor  in  series  with  a  ballistic  galvanometer 
(p.  156).  The  earth  inductor  is  first  placed  with  the  plane 
of  its  coils  vertical  and  perpendicular  to  the  magnetic 
meridian.  It  is  then  rotated  through  180°  and  the  throw  d^ 
noted.  Several  readings  should  be  made.  The  plane  of 
the  coils  is  then  placed  horizontally  and  the  throw  d2  on 
rotation  through  180°  noted.  The  ratio  of  d,  to  dl  is  the 
tangent  of  the  dip. 

Questions. 

1.  What  other  sources  of  error  may  there  be  in  measurement  by 
the  dip-circle? 

2.  Would  you  be  justified  in  making  a  calculation  of  "probable 
error"  from  the  various  readings  with  the  dip-circle? 

3.  If  you  have  performed  Exp.  XLII,  calculate   the   total   force 
and  the  vertical  component. 

XLIV.  MEASUREMENT    OF    RESISTANCE    BY    WHEAT- 
STONE'S  BRIDGE. 

Ames'  General  Physics,  pp.  725-727;  Watson's  Physics,  pp.  685-687; 
Text-book  of  Physics  (Duff),  p.  572;  Hadley's  Electricity  and 
Magnetism,  pp.  306-310;  Watson's  Practical  Physics,  pp.  432- 
437;  Kohlrausch's  Physical  Measurements,  p.  303. 

The  practical  measurement  of  a  resistance  consists  in 
comparing  it  with  a  known  or  standard  resistance.  For 


170  ELECTRICITY  AND    MAGNETISM. 

resistances  of  medium  magnitude,  Wheatstone's  Bridge  is 
usually  used  (p.  153). 

In  joining  the  known  and  unknown  resistances  to  the 
bridge,  connectors  should  be  used  whose  resistance  is  negli- 
gible; that  is,  less  than  the  unavoidable  error  that  may 
occur  in  determining  the  unknown  resistance.  In  con- 
necting the  battery  and  galvanometer,  no  such  precaution  is 
necessary,  for  their  resistances  do  not  enter  into  the  calcu- 
lation. The  galvanometer  may  be  connected  to  either  pair 
of  opposite  corners;  but,  where  the  greatest  sensitiveness 
is  required,  if  the  galvanometer  has  a  higher  resistance 
than  the  battery,  it  should  be  in  the  branch  that  connects 
the  junction  of  the  highest  two  of  the  four  resistances  P, 
Q,  R,  S  to  the  junction  of  the  lowest  two;  while,  if  the 
battery  has  the  greatest  resistance,  it  should  occupy  that 
position.  Two  spring  keys  should  be  included  in  the  con- 
nections, one  in  the  battery  arm  and  the  other  in  the  gal- 
vanometer arm.  When  testing  for  a  balance,  the  battery 
key  should  be  pressed  first,  then  the  galvanometer  key. 
If  taken  in  the  reverse  order,  there  might  be  a  small  deflec- 
tion due  to  the  self-induction  of  the  various  parts.  These 
keys  should  be  pressed  for  a  moment  only.  Except  for  a 
final  determination,  it  is  not  necessary  to  wait  until  the  gal- 
vanometer has  quite  come  to  rest,  for  a  lack  of  balance  will 
be  indicated  by  a  sudden  disturbance  of  the  swing  when 
the  galvanometer  key  is  pressed.  The  pressure  of  the  gal- 
vanometer key  should  be  brief,  sufficient  merely  to  indicate 
the  direction  of  the  initial  movement. 

In  practice,  it  is  best  to  use  a  box-resistance  as  nearly 
as  possible  equal  to  the  unknown  resistance.  This  comes 
to  the  same  thing  as  saying  that  the  box-resistance  should 
be  varied  until  a  balance  is  attained  when  the  parts  of  the 
meter  wire  are  nearly  equal.  The  reason  for  this  preference 
is  that  the  sensitiveness  is  then  a  maximum,  or  a  slight  lack 
of  balance  is  most  easily  detected  by  the  deflection  of  the 
galvanometer.  The  exact  ratio  of  P  to  Q  for  a  balance 
should  be  very  carefully  ascertained.  At  least  six  settings 


GALVANOMETER    RESISTANCE    BY    SHUNT    METHOD.       171 

should  be  made ;  and  to  secure  independence  of  the  settings, 
the  eye  should  be  kept  on  the  galvanometer-scale  and  the 
reading  of  the  bridge  not  examined  until  the  setting  has 
been  decided  on.  The  mean  of  these  six  is  then  taken.  R 
and  5  should  then  be  interchanged  and  six  more  settings 
made.  This  interchange  will  serve  to  eliminate  the  effect 
of  lack  of  symmetry  of  the  two  sides  of  the  wire  bridge 
and  its  connections. 

The  structure  of  the  galvanometer  to  be  used,  its  coils, 
magnets,  and  connections,  should  be  carefully  examined  and 
care  taken  that  it  is  thoroughly  understood  (p.  155). 

Three  unknown  resistances  should  be  measured  sep- 
arately and  then  all  in  parallel.  From  the  separate  resist- 
ances the  resistance  of  the  conductors  in  parallel  should  be 
calculated  and  compared  with  the  measurement  of  the 
same.  The  resistance  of  a  wire  should  then  be  measured 
and  its  length  and  mean  diameter  obtained.  From  these 
data,  the  specific  resistance  of  the  material  of  the  wire 
should  be  deduced.  The  temperature  at  which  the  resist- 
ance is  measured  should  also  be  noted,  and  from  the  tem- 
perature coefficient  of  the  material  (Table  XXII)  the  specific 
resistance  at  o°  C.  calculated. 

The  possible  errors  of  the  measurements,  and  hence  the 
extent  to  which  the  calculations  should  be  carried,  may  be 
deduced  from  the  mean  deviation  in  each  set  of  readings. 

Questions. 

1 .  Does  the  battery  need  to  be  a  constant  one  ? 

2.  What  objections  are  there  to  allowing  the  battery  circuit  to 
remain  closed? 

3.  Why  is  it  difficult  by  this  method  to  measure  very  large  or  very 
small  resistances  ? 

XLV.  GALVANOMETER    RESISTANCE    BY   SHUNT 
METHOD. 

Kohlrausch's  Physical  Measurements,  p.  325. 
If   a   galvanometer   of  resistance   G   connected   in   series 
with  a  battery  of  resistance  B  and  e.  m.  f.  E  and  a  box 
resistance  R  gives  a  deflection  d  and  if  C  be  the  current 


172 


ELECTRICITY    AND    MAGNETISM. 


C  = 


R+B+G 


=  Kd 


where  K  is  a  constant  for  the  galvanometer.  If  now  the 
galvanometer  be  shunted  by  a  resistance  S  and  the  deflection 
be  then  df  and  the  current  through  the  galvanometer  C', 

E  S 


/-*/ 


R+B  + 


GS   XG+S 


=  Kd'. 


G  +  S 


Hence 


(R+B)  (G+S)+GS_d 
S  (R+B+G)        =~dr> 

and  from  this  G  is  readily  deduced  provided   B  is  known. 
Usually  a  battery  of  such  low  resistance  can  be  used  that  B 
is  negligible  compared  with  R  and  may  be  omitted;  otherwise 
A        ^  B    must    be    obtained    as   in   Exp.    LI  II. 

The  galvanometer  should  be  connected 
through  a  commutator  and  several  read- 
ings on  both  sides  should  be  made. 

If  the  e.  m.  f.  of  the  cell  supplied  is  too 
great,  a  suitable  fraction  of  it  should  be 
employed  (p.  161). 

As  a  check,  the  determination  of  G 
should  be  repeated,  a  different  value  for 
5  being  used.  If  the  galvanometer  is 
very  sensitive,  its  resistance  must  be  found 
from  two  readings  with  shunts.  A  suita- 
ble formula  is  readily  worked  out. 

If  R  should  be  very  great  compared  with  the  other 
resistances,  the  formula  may  be  simplified.  This  will 
usually  be  the  case  if  the  galvanometer  is  very  sensitive  or  of 
low  resistance.  The  quantities  added  to  R  in  the  first  two 
equations  may  then  be  neglected  and  we  get 


G+S     d 


GALVONOMETER  RESISTANCE  BY  THOMSON'S  METHOD.       173 


XLVI.    GALVANOMETER    RESISTANCE    BY    THOMSON'S 

METHOD. 

Hadley's  Electricity  and  Magnetism,  p.  321;  Kohlrausch's  Physica 
Measurements,  p.  328;  Stewart  and  Gee's  Practical  Physics,  II, 
p.  140-142. 

The  resistance  of  the  coils  of  a  galvanometer  may  be  found 
by  means  of  Wheatstone's  Bridge  as  the  resistance  of  any 
ordinary  conductor  is  found.  This  would  require  the  use 
of  a  second  galvanometer.  The  second  galvanometer,  for 
detecting  when  the  bridge  is  balanced,  is  frequently  un- 
necessary. The  condition  for 
a  balance  is  that,  when  the 
branch  in  which  the  galvanom- 
eter is  usually  placed  is  closed 
by  a  key,  no  current  shall  flow 
through  it.  If  a  current  did 
flow  through  it,  a  change 
would  take  place  in  the  cur- 
rents in  the  other  arms.  Now 
the  presence  of  a  galvanom- 
eter in  one  of  these  other  arms 
enables  us  to  test  whether  any 
change  in  the  distribution  of 
the  currents  takes  place  on 

the  key's  being  pressed.  Hence,  in  Thomson's  method  for 
galvanometer  resistance  the  galvanometer  is  placed  in  the 
"unknown"  arm  and  a  spring  key,  K,  is  placed  in  the 
branch  in  which,  in  the  ordinary  arrangement  of  Wheat- 
stone's  Bridge,  a  galvanometer  is  found.  A  diagram  to 
illustrate  the  connections  is  given  in  figure  50. 

From  the  above  it  will  be  seen  that  in  this  method  a  bal- 
ance is  obtained  when  the  deflection  of  the  galvanometer 
does  not  change  on  the  key,  K,  being  pressed.  Two  practical 
difficulties  are  met  with.  The  first  is  that  the  deflection 
of  the  galvanometer  before  the  key  is  pressed  may  be  so 
large  that  it  cannot  be  read.  When  the  galvanometer  is  of 


FIG.  50. 


174  ELECTRICITY   AND    MAGNETISM. 

the  Thomson  type  (p.  155),  this  difficulty  may  be  overcome 
by  turning  the  control  magnet  until  the  deflection  can  be 
read  (the  zero  position  of  the  galvanometer  could,  of  course, 
not  then  be  read  on  the  scale,  but  that  is  not  necessary). 
In  the  d'Arsonval  type  of  galvanometer  there  is  no  such 
way  of  overcoming  this  difficulty,  and  so  this  method  is  not 
so  easily  applied  to  such  a  galvanometer.  The  second 
difficulty  is  that  if  the  battery  be  a  variable  one,  the  gal- 
vanometer will  not  give  a  steady  deflection.  Hence,  a 
constant  battery  of  the  Daniell  or  Gravity  type  should  be 
used  (p.  159).  It  may  also  be  necessary  to  decrease  the 
current  through  the  bridge  and  galvanometer  by  putting 
considerable  resistance  in  series  with  the  battery,  or  a  fraction 
of  the  e.  m.  f.  of  the  cell  may  be  used  (p.  161). 

In  the  experiment  it  is  better  to  use  a  bridge-box  instead 
of  a  wire  bridge,  for  the  condition  for  sensitiveness,  that 
the  arms  should  be  as  nearly  equal  as  possible,  still  holds, 
and  the  resistance  of  a  wire  bridge  is  usually  very  small 
compared  with  that  of  the  galvanometer.  Beginners  some- 
times find  difficulty  in  deciding  on  the  proper  connections. 
The  best  way  is  to  consider  what  the  connections  would  be 
in  the  ordinary  use  of  Wheatstone's  Bridge,  and  then  con- 
sider the  modifications  introduced  in  the  present  method. 
If  possible,  ratio  arms  of  1000  to  1000,  100  to  1000,  and  10 
to  1000  should  be  used  in  succession  to  obtain  successive 
approximations.  The  last  should  give  the  resistance  to  two 
places  of  decimals  (if  one  ohm  is  the  least  box-resistance), 
but  the  decreasing  sensitiveness  may  prevent  the  latter 
ratios  from  giving  more  accurate  results  than  the  first. 

If  the  galvanometer  has  more  than  one  coil,  the  resist- 
ance of  each  should  be  measured  separately  and  then  the 
resistance  of  all  in  series.  This  will  afford  a  check  on  the 
work. 

Question. 

i.  Describe  carefully  the  swinging  system,  coils,  control  magnet, 
and  connections  of  the  galvanometer  used. 


MEASUREMENT    OF    HIGH    RESISTANCES.  175 

XLVII.  MEASUREMENT  OF  HIGH  RESISTANCES  (i). 

Watson's   Practical   Physics,    pp.    460-461;    Henderson's   Electricity 
and  Magnetism,  pp.  66-72. 

The  method  of  Wheatstone's  Bridge  is  not  suitable  for 
measuring  very  high  resistances.  One  method  is  to  con- 
nect the  unknown  resistance  X,  a  battery  of  negligible  re- 
sistance and  e.  m.  f.  E,  and  a  sensitive  galvanometer  of 
resistance  G  in  series.  If  the  current  be  C, 

E 

C  =  -  —  ,  giving  a  deflection  d. 
X  -\-G 

Now  replace  X  by  a  known  resistance,  R,  and  shunt 
the  galvanometer  by  such  a  resistance,  5,  that  the  deflection 
is  readable.  By  considering  the  total  current  and  the  part 
C'  of  the  total  current  that  passes  through  the  galvanometer, 
we  readily  find  that 

W  S" 

C'  —  ---  ,  giving  a  deflection  d'  , 
GS   G+S 

+ 


Hence, 

R(G+S)+GS  _d 
S(X+G)      =J" 

and  from  this  X  is  readily  deduced.  G  may  be  found  as  in 
Exp.  XLV  or  XLVI;  but  if  (G+S)/S  is  known  and  G  is 
small  compared  with  X  and  R,  the  resistance  of  the  galvan- 
ometer need  not  be  determined.  Many  galvanometers  are 
provided  with  shunt  boxes,  for  which  S/(G+S)  is  o.i, 
o.oi,  or  o.ooi. 

The  galvanometer  should  be  connected  through  a  com- 
mutator, and  several  readings  on  both  sides  should  be  made 
to  obtain  a  reliable  mean. 

As  a  check,  repeat  the  measurements  with  a  different 
value  for  R  and  a  different  value  for  S. 


76 


ELECTRICITY   AND    MAGNETISM. 


XLVIII.  MEASUREMENT    OF    HIGH    RESISTANCES    (2). 

Text-book  of  Physics  (Duff),  pp.  537-538;  Watson's  Physics,  pp. 
656-657;  Ames'  General  Physics,  pp.  658-659;  Henderson's 
Electricity  and  Magnetism,  pp.  71-75;  Hadley's  Electricity  and 
Magnetism,  pp.  202-210;  Watson's  Practical  Physics,  pp.  569- 
571- 

A  very  high  resistance,  such  as  the  insulation  resistance  of 
a  cable  or  the  resistance  of  cloth,  paper,  wood,  etc.,  may 
be  measured  by  finding  the  rate  at  which  the  electricity  in  a 
charged  condenser  leaks  through  the  conductor.  An 
electrometer  is  used  to  find  the  change  of  potential  of  the 
condenser  and  from  this  the  rate  of  loss  of  its  charge  is 
deduced.  The  Dolezalek  form  of  Kelvin's  quadrant  electrom- 


FIG.  51. 

eter  is  suitable.  Its  needle  is  kept  charged  to  a  high  poten- 
tial by  being  connected  to  one  pole  of  a  battery  of  small 
cells,  the  other  pole  being  grounded. 

To  find  the  insulation  resistance  of  a  cable  the  whole  of 
the  cable  except  the  ends  is  immersed  in  a  tank  of  salt  water 
which  is  connected  to  the  earth.  One  of  the  ends  is  carefully 
paraffined  to  prevent  surface  leakage  and  the  core  of  the 
other  end  is  connected  to  the  insulated  pair  of  quadrants. 
If  the  cable  is  sheathed  with  metal,  immersion  is  not  neces- 
sary. Other  materials,  such  as  those  mentioned,  are 
pressed  between  sheets  of  tinfoil,  one  sheet  being  connected 
to  the  earthed  quadrants  and  the  other  to  the  insulated 
quadrants. 


MEASUREMENT    OF   HIGH    RESISTANCES.  177 

Let  V\  =  potential  given  the  condenser  on  closing  the 
key  K.  The  charge  Q  in  the  condenser  and  cable  =  C  V^ 
where  C  is  their  joint  capacity.  Upon  opening  K  the  charge 
flows  through  a  resistance  R±  for  a  time,  t,  ^t  being  the  insul- 
ation resistance  of  the  cable,  the  condenser,  and  the  elec- 
trometer and  keys  in  parallel. 

Since  the  current  at  the  time  /  equals  V/R^  by  Ohm's  law, 
and  also  equals  the  rate  of  decrease  of  Q  or  of  CV 

—  =  -cdv- 

R,  dt 

dV  _dt 

'T".s; 

Integrating  between  the  limits  t  =  o  when   V=V1  and  /  =  t 
when  V=V2  we  get 


R  t          ^°-434* 

C  l°ge — ~       C  1°£~  * 

where  dl  and  d2  are  the  initial  and  final  deflections  of  the 
electrometer  from  the  zero  position. 

The  zero  should  be  determined  both  before  Vl  is  found 
and  after  V2  is  found.  As  it  is  very  apt  to  vary  slightly, 
more  reliable  results  can  be  attained  by  continuing  to  read 
V  at  intervals  (e.  g.,  every  half -minute)  until  it  has  fallen 
to  about  one-half  of  its  original  value.  From  a  curve  drawn 
to  represent  V  and  /,  two  reliable  points  may  be  chosen 
to  give  values  for  Vl  and  V2  to  be  used  in  the  calculation. 

A  subdivided  condenser  is  desirable  in  order  that  a 
capacity  giving  a  sufficiently  rapid  fall  of  potential  may  be 
chosen. 

The  total  insulation  resistance,  R2)  of  the  other  parts  in 
parallel  with  the  cable  are  found  by  disconnecting  the 
cable  and  making  a  second  set  of  observations  as  above. 

12 


178  ELECTRICITY  AND   MAGNETISM. 

The  insulation  resistance,  R,  of  the  cable  is  then  deducible 
for 


The  capacity,  Clt  of  the  cable  can  be  compared  with 
that  of  the  condenser,  C2,  by  the  method  of  "divided 
charge."  First  charge  the  condenser  and  observe  its 
potential  by  the  electrometer  and  let  the  deflection  be  dr 
Then  connect  in  the  cable  and  let  d2  be  the  new  deflection. 
Since  the  total  charge  Q  remains  unchanged, 


and,  since  the  deflections  are  proportional  to  the  potentials, 


C=Cl       2 

Questions. 

1.  Why  should  one  pole  of  the  battery  that  charges  the  needle  be 
grounded  ? 

2.  Why  must  keys  of  specially  high  insulation  be  used  in  this 
method  ? 

3.  Calculate  the  capacity  of  the  cable  in  electrostatic  units  from 
rough  measurements  of  its  dimensions  and  reduce  to  microfarads 
(see  p.  162). 

XLIX.  MEASUREMENT  OF  LOW  RESISTANCES  (i). 

Very  low  resistances  cannot  be  measured  by  the  Wheatstone 
Bridge  method,  because  the  unknown  resistances  of  the 
connections  are  not  small  compared  with  the  resistance 
to  be  measured.  The  simplest  method  for  low  resistances 
is  a  "fall  of  potential  "  method.  A  current  is  passed  through 
the  resistance,  the  current  is  measured  by  an  ammeter  and 
the  difference  of  potential  is  measured  by  a  voltmeter;  then 
the  resistance  is  known  from  Ohm's  Law.  For  very  low 
resistances  the  fall  of  potential  will  be  very  small  and  an 
instrument  much  more  sensitive  than  any  commercial  volt- 
meter must  be  used.  Instead  of  a  voltmeter  a  sensitive  gal- 
vanometer of  high  resistance,  or  a  low  resistance  galvan- 
ometer in  series  with  a  high  resistance,  is  used  and  the 


MEASUREMENT    OF    LOW    RESISTANCES.  179 

value  of  a  scale  division  of  the  galvanometer  regarded  as  a 
voltmeter  is  found  by  a  separate  experiment. 

Let  the  resistance  to  be  measured  be  x,  and  let  the  differ- 
ence of  potential  at  its  ends  when  current  C  passes  through 
it  be  e.  Then 

e 

x=c 

C,  which  should  be  large,  may  be  measured  by  an  ammeter. 
To  find  e  we  must  know  the  constant,  K,  of  the  galvanometer 
considered  as  a  voltmeter;  that  is,  the  number  of  volts  per 
unit  deflection.  If  the  deflection  is  D 

e=K.D 

To  find  K  apply  to  the  galvanometer  a  small  fraction  of 
the  e.  m.  f.,  E,  of  a  Daniell'scell  (p.  159).  ,  E 

For  this  purpose  connect  the  cell  in 
series  with  a  very  high  resistance  box 
and  a  box  of  moderate  resistances  and 
join  the  galvanometer  to  the  ends  of 
one  of  the  small  resistances,  r,  choosing 
r  so  that  the  deflection,  d,  will  not  be 
very  different  from  D.  Then  if  the  re- 
sistance of  the  galvanometer  be  great 
compared  with  r  (see  p.  161)  and 
if  the  total  resistance  in  series  with  FIG.  52. 

the  battery  be  R,  the  e.  m.  f.  acting  on  the  galvanometer 
is  Er/R. 
Hence 


As  a  check  on  the  work  redetermine  K  using  a  different 
value  of  r.     From  the  above  equations  x  is  found. 

In  the  first  part  of  the  experiment  place  a  commutator 
in  the  main  circuit  so  that  C  may  be  reversed  and  the  effect 
of  thermo-electric  forces  at  the  contacts  eliminated,  and 
connect  the  galvanometer  through  a  second  commutator  so 
that  lack  of  symmetry  in  its  deflection  may  be  eliminated. 


i8o 


ELECTRICITY   AND    MAGNETISM. 


Thus  D  will  be  the  mean  of  four  readings.  Exactly  simi- 
lar precautions  should  be  observed  in  the  second  part. 
Close  the  currents  only  for  the  shortest  possible  times 
necessary  to  make  the  readings,  otherwise  heating  may 
occur  and  resistances  (especially  the  unknown  x)  may  change. 
The  determination  should  be  repeated  several  times  with 
different  values  of  C.  If  the  work  has  been  reliable,  D  should 
be  proportional  to  C.  Note  also  the  temperature  of  the 
specimen  and  calculate  its  resistivity  from  its  resistance  and 
dimensions. 

Questions. 

1.  If  r  had  not  been  negligible  compared  with  the  galvanometer 
resistance  how  would  this  have  appeared  in  the  course  of  the  work? 

2.  Find  the  equation  that  must  replace  the  above  if  the  resist- 
ance of  the  battery  is  not  negligible  compared  with  R  and  if  r  is  not 
negligible  compared  with  the  galvanometer  resistance. 

L.  MEASUREMENT  OF  LOW  RESISTANCES  (2). 

Henderson's  Electricity  and  Magnetism,  pp.  57—58;  Stewart  and  Gee's 
Practical  Physics,  II,  pp.  177-181. 

When  a  standard  low  resistance  (o.oi   or  o.ooi   ohm)  is 
available,  a  conductor  of  low  resistance  x  may  be  connected 

in  series  with  it  and  a  battery, 
and  a  very  sensitive  voltmeter, 
or  a  high-resistance  galvanom- 
eter, serving  as  a  voltmeter, 
may  be  used  to  compare  the 
falls  of  potential  in  x  and  the 
standard.  The  resistances  will 
be  proportional  to  the  falls  of 
potential. 

Connection  with  the  battery 
should  be  made  through  a  com- 
mutator to  reverse  thermal 
effects  at  the  connections,  and 
the  galvanometer  should  be 
connected  through  a  second 
commutator  to  eliminate  asym- 


MEASUREMENT    OF    LOW    RESISTANCES. 


181 


metry  of  the  galvanometer  readings.     Thus  each  final  read- 
ing will  be  the  mean  of  four  separate  readings. 

The  currents  should  be  closed  for  the  shortest  times 
sufficient  for  the  readings,  to  avoid  heating.  Note  the 
temperature  of  the  specimen. 

Questions. 

1.  What  are  the  comparative  advantages  and  disadvantages  of  this 
and  the  preceding  method  ? 

2.  Why  is  a  high-resistance  galvanometer  to  be  preferred? 

3.  Will  poor  contact  have  as  much  effect  as  in  a  measurement  of 
low  resistance  by  Wheatstone's  Bridge?     Why? 


LI.  MEASUREMENT    OF    LOW    RESISTANCES    BY    THE 
THOMSON  DOUBLE  BRIDGE. 

Watson's  Practical  Physics,  pp.  465-469;  Stewart  and  Gee,  II, 
pp. 182-187. 

In  Thomson's  Double  Bridge  the  errors  of  the  contacts 
in  the  use  of  Wheatstone's  Bridge  are  avoided.  Its  princi- 
ple is,  in  fact,  that  of  the  fall  of  potential  method  (Exp. 
L)  the  direct  comparison  of  the  falls 
of  potential  being  replaced  by  a  null 
method.  This  method  is  applicable  to 
extremely  low  resistances  as  well  as  to 
medium  resistances. 

In  the  diagram  x  is  the  resistance  to 
be  measured  and  r  a  standard  known 
resistance;  a  may  be  made  10  or  100;   " 
and  b  may  be  made  100,  1,000,  10,000. 
Similarly,  a'  may  be  made   10  or  100 


and  bf  100,  1,000,  10,000.     Now  it  can 


FIG.  54. 


be  shown  that  if  there  is  no  current  in  the  galvanometer,  and 
if  the  ratios  a/  b  and  a'/b'  are  equal, 

x     a      a' 

~r=b=l)'' 

hence  a  value  of  r  (which  is  variable)  and  values  of  a,  b, 
a'  b',  are  sought  which  give  no  current  in  the  galvanometer, 
and  from  these  x  is  calculated. 


l82 


ELECTRICITY  AND   MAGNETISM. 


The  form  of  Double  Bridge  made  by  Hartmann  and  Braun 
is  very  satisfactory.  The  correspondence  of  parts  to  parts 
of  the  diagram  is  readily  traced.  The  ratio  of  a  to  b  can  be 
varied  from  100  to  100  to  10  to  10,000;  moreover,  a  and  b 
may  be  interchanged  and  so  the  ratio  reversed.  Similar  re- 
marks apply  to  a'  and  b'.  Thus  values  of  x/r  varying 
from  10/10000  to  10000/10  may  be  measured.  The 
variable  r  may  be  varied  from  o .  044  down  to  o,  but  can 
hardly  be  read  with  an  accuracy  of  i%  below  o.ooi. 

Hence  values  of  x  between 
o.ooi/ 1000  or  o.oooooi  and 
0.044X1000  or  44,  may  be 
measured  by  the  bridge. 

Care  must  be  taken  not  to 
injure  the  standardized  bar  by 
scraping  the  contact  maker  along 
it.  The  contact  maker  must  be 
raised  for  each  movement.  Do 
not  allow  the  sharp  jaws  of  the 
clamps  to  come  down  on  the  bar 
too  suddenly,  for  they  might  cut 
into  the  bar  somewhat. 

Test  as  many  as  possible  of  the  following  materials: 
(i)    Brass.      (2)   Iron.      (3)   Copper.      (4)   Zinc.      (5)   Lead. 
(6)   Carbon.     (7)   Rail  Bond, 
and  calculate  the  Specific  Resistance  of  each. 

Proof  of  Formula. 

Regard  Thomson's  Double  Bridge  as  a  modified  Wheatstone's 
Bridge,  the  modification  consisting  in  the  paralleling  of  parts  of  the 
arms  AB  and  BC  as  indicated.  Let  G  and  D  be  points  at  the  same 
potential  as  indicated  by  the  galvanometer.  E  B  F  is  the  heavy 
conductor  joining  the  unknown  and  the  standard.  Let  B  be  a 
point  in  it  at  the  same  potential  as  G.  We  may  suppose  B  and  G 
permanently  connected.  Let  the  resistance  of  p  and  a'  in  parallel 
be  m,  and  that  of  q  and  b'  be  n.  Then,  by  the  Wheatstone  Bridge 
formula : 


FIG.  55. 


a  _x  +  m 
b~  r  +  n' 


COMPARISON    OF    RESISTANCES.  183 

Now  if  we  show  that 


it  will  follow  that 

a  =  x_ 
b     7' 

To  show  this,  note  that 

a  _  a'  _  p 


a'  +  p    b'+q' 

Now 

=  a'p         ^b^q 

m    p     a 

Hence 

x     a 

Questions. 

1.  Considering  this  as  a  modified  fall   of   potential   method,  why 
should  a,  b,  a',  b',  be  of  very  large  and  E  B  F  of  very  small  resistance? 

2.  Does  the  battery  current  need  to  be  steady?     Why? 

3.  Could  an  alternating  current  be  used  in  any  circumstances? 

LII.  COMPARISON  OF  RESISTANCES  BY  THE   CAREY- 
FOSTER  METHOD. 

Watson's  Practical  Physics,  pp.  442-446;  Schuster  and  Lees'  Practical 
Physics,  pp.  307-309;  Henderson's  Electricity  and  Magnetism, 
pp.  53—57;  Stewart  and  Gee's  Practical  Physics,  II,  pp.  158—170. 

To  find  very  accurately  the  difference  between  two  very 
nearly  equal  resistances  R  and  5,  connect  them  and  two 
other  nearly  equal  resistances,  P  and  Q,  as  indicated  in  the 
diagram,  where  ab  is  a  very  uniform  wire,  which  we  shall 
suppose  to  have  a  resistance  of  more  than  i  ohm.  Let  the 
resistance  of  unit  length  of  the  wire  ab  be  p.  Let  the  dis- 
tance ad  be  xlt  when  a  balance  has  been  obtained  in  the 
usual  way.  Then  exchange  R  and  5,  and  again  obtain  a 
balance.  Denote  the  new  value  of  ad  by  x2.  Since  P  and  Q 


1 84 


ELECTRICITY   AND    MAGNETISM. 


have  not  been  changed  and  the  total  resistance  R,  S,  and  ab 
was  not  changed,  it  is  clear  that  R  -i-x1p  =  S  +x^p,  or 
R  —  S=  (x2  —  xjp.  To  find  the  value  of  p,  replace  R  by  a 
standard  i-ohm  coil,  and  5  by  a  heavy  connector  of  neg- 


FlG. 


ligible  resistance,  and  proceed  as  above;  then  p(x2  —  xl)  =  i. 
The  exchange  of  R  and  5  is  made  by  means  of  a  special 
key  designed  so  that  the  resistance  of  the  connections  will 
remain  the  same  (see  Fig.  57). 


FIG.  57. 

To  compare  a  box  of  unknown  errors  with  a  standardized 
box,  the  difference  between  each  resistance  of  the  former 
and  a  corresponding  resistance  of  the  latter  is  found  by  the 
above  method. 


BATTERY    RESISTANCE    BY    MANGE'S    METHOD. 


185 


To  calibrate  two  boxes,  put  one  in  place  of  R  and  replace 
5  by  a  standard  i-ohm  coil  and  so  find  exactly  the  value 
of  each  i-ohm  unit  in  the  box.  Then  replace  the  standard 
by  the  other  box,  in  position  S,  and  compare  the  i-ohm 
units  of  the  second  box  with  those  of  the  first  box.  Then 
compare  a  2-ohm  unit  in  one  box  with  two  i-ohms  in 
the  other,  and  so  on.  Special  care  must  be  taken  to  avoid 
confusion  in  making  the  calculations,  and  for  this  purpose 
the  box  resistances  may  be  denoted  by  Ilf  I2,  II  lt  II2,  etc., 
for  one  box,  and  I\,  I'2,  II'2,  etc.,  for  the  other. 

Questions. 

1.  State  the  formula  for  Wheatstone's  Bridge  before  and  after  R 
and  5"  are  interchanged  and  therefrom  deduce  the  above  formula. 

2.  Do  P  and  Q  need  to  be  exactly  equal,  and  why? 


LIII.  BATTERY  RESISTANCE  BY  MANGE'S  METHOD. 

Hartley's  Electricity  and  Magnetism,  p.  322 ;  Watson's  Practical  Physics, 
p.  475;  Schuster  and  Lee's  Practical  Physics,  pp.  303-306. 

The  resistance  of  a  battery  may  be  determined  by  plac- 
ing it  in  the  "unknown  arm"  .R  of  a  Wheatstone's  Bridge 
(p-  I53)-  In  "this  case  there 
will  be  a  current  through 
the  galvanometer  when  the 
bridge  battery  is  not  con- 
nected. But  if  P,  Q  and  5 
be  adjusted  until  there  is  no 
change  in  the  deflection  when 
the  key  of  the  bridge  battery 
is  pressed,  the  points  to 
which  the  galvanometer  is 
connected  will  be  at  the  same 
potential  so  far  as  the  effect 
of  the  bridge  battery  is  con- 
cerned. Since,  when  the 


FIG.  58. 


adjustments  are  right,  the  bridge  battery  sends  no  current 
through  the  galvanometer,  this  battery  may  be  removed  and 
the  key  alone  will  serve  to  test  the  adjustment  of  P,  Q,  and  S. 


1 86  ELECTRICITY  AND   MAGNETISM. 

If  the  deflection  of  the  galvanometer  is  too  great  to  be 
readable,  the  control  magnet  (in  the  case  of  a  Kelvin  gal- 
vanometer) may  be  used  to  bring  the  needle  back,  or  the 
galvanometer  may  be  shunted  or  a  resistance  put  in  series 
with  it.  Most  cells  vary  slightly  in  resistance  and  e.  m.  f. 
when  on  closed  circuit;  hence,  the  keys  should  not  be 
pressed  longer  than  is  necessary. 

In  Lodge's  modification  of  Mance's  Method  a  condenser 
is  placed  in  series  with  the  galvanometer.  There  will  then 
be  no  continuous  current  through  the  galvanometer;  but,  if 
the  adjustments  of  P,  Q,  and  5  are  not  right,  on  pressing 
the  key  by  which  the  adjustment  is  tested  the  galvanometer 
will  be  momentarily  deflected. 

Questions. 

1.  Why  is  there  a  slow  movement  of  the  galvanometer  needle 
when  the  keys  are  kept  pressed  ? 

2.  Should  the  condenser  be  of  large  or  small  capacity?     Would  a 
Ley  den  jar  do? 

LIV.  TEMPERATURE    COEFFICIENT    OF    RESISTANCE. 

Text-book  of  Physics  (Duff),  p.  564-566;  Ames'  General  Physics,  pp. 
731-732;  Watson's  Physics,  pp.  681-682;  Hadley's  Electricity 
and  Magnetism,  p.  294;  Henderson's  Electricity  and  Magnetism, 
pp.  95-101. 

The  resistance  of  most  solids  increases  as  the  tempera- 
ture rises;  carbon  is  one  of  the  exceptions,  for  its  resistance 
decreases.  For  moderate  ranges  of  temperature  the  resist- 
ance is  approximately  a  linear  function  of  the  temperature  or, 
if  ^0  be  the  resistance  at  o°  and  R  that  at  t°, 

R=RQ(i+at) 

The  constant  a  is  called  the  temperature  coefficient  of  the  ma- 
terial. It  may  be  defined  as  the  change  per  ohm,  referred 
to  the  resistance  at  o°,  per  degree  change  of  temperature. 

The  change  of  resistance  can  be  most  conveniently  studied 
by  the  box  form  of  Wheatstone's  Bridge  (p.  154). 

(A)  For  finding  the  temperature  coefficient  of  a  wire  such 
as  copper,  a  length  sufficient  to  give  several  ohms  resistance 


TEMPERATURE    COEFFICIENT    OF    RESISTANCE.  187 

should  be  used.  The  determination  of  the  temperature 
coefficient  does  not  require  that  the  dimensions  of  the 
specimen  should  be  known,  but  the  specific  resistance  of 
the  specimen  might  as  well  be  determined  at  the  same  time. 
Hence  the  length  and  mean  diameter  of  the  wire  should  be 
carefully  measured.  The  wire  should  then  be  soldered  to 
heavier  lead  wires  and  immersed  in  a  bath  of  oil,  and  its 
resistance  determined  at  intervals  of  about  10°  as  the  tem- 
perature is  raised.  The  thermometer  should  be  placed  inside 
the  coil  so  as  to  be  as  nearly  as  possible  at  the  temperature 
of  the  latter.  It  will  be  an  improvement  if  the  coil  and 
thermometer  are  in  a  test-tube  that  is  immersed  in  the 
bath,  the  opening  of  the  tube  being  closed  with  cotton-wool. 

To  keep  the  temperature  constant,  while  measuring  the 
resistance,  would  be  difficult.  The  following  method  will 
be  found  to  give  much  better  results:  Having  measured 
the  resistance  at  the  temperature  of  the  room,  adjust  the 
known  resistance  of  the  bridge  so  that  there  would  be  a 
balance  if  the  resistance  of  the  wire  were  increased  4  or  5 
per  cent.  The  galvanometer  will  be  deflected.  Now  heat 
the  wire  very  slowly  and  the  galvanometer  reading  will 
begin  to  drift  toward  zero.  When  it  just  reaches  zero,  read 
the  thermometer  and  continue  the  process  step  by  step. 

The  various  resistances  and  temperatures  should  then 
be  plotted  in  a  curve  that  should  be  approximately  a  straight 
line.  If  exactly  a  straight  line  is  obtained,  the  temperature 
coefficient  should  be  calculated  from  two  reliable  and  widely 
separated  points  on  the  curve.  Let  R  and  R'  be  the  resist- 
ances at  t  and  t',  respectively.  Substituting  these  values  in 
the  above  equation  we  shall  get  two  equations  from  which 
R0  can  be  eliminated. 

If  the  plotted  readings  give  a  distinct  curve,  the  resistance 
must  be  expressed  as  a  quadratic  function  of  the  temperature. 

R=RQ(I  +  at  +  bt2) 

From  three  points  on  the  curve  three  equations  may  be 
written  down  and  from  these  a  and  b  may  be  calculated. 


1 88  ELECTRICITY   AND    MAGNETISM. 

(B)  For  finding  the  temperature  coefficient  of  carbon 
an  incandescent  lamp  may  be  used.  As  it  would  be  diffi- 
cult to  determine  accurately  the  temperature  of  the  filament 
in  the  exhausted  bulb  by  the  preceding  method,  water  may 
be  used  for  the  bath  and  two  careful  determinations  of  the 
resistance  made,  the  first  being  while  the  water  is  at  about 
the  temperature  of  the  room,  and  the  other  when  the  water 
is  boiling.  In  each  case  the  final  determination  of  the 
resistance  should  not  be  made  until  the  temperature  of  the 
filament  has  become  constant,  as  is  indicated  by  its  resist- 
ance becoming  quite  constant.  The  leads,  where  they  are 
immersed  in  the  water,  should  be  carefully  insulated  with 
tape. 

LV.  SPECIFIC  RESISTANCE  OF  AN  ELECTROLYTE. 

Watson's  Practical  Physics,  pp.  475-486;  Henderson's  Electricity  and 
Magnetism,  pp.  80—84;  EwelVs  Physical  Chemistry,  pp.  54-57; 
Kohlrausch's  Physical  Measurements,  pp.  316-321. 

The  object  of  this  experiment  is  to  determine  the  specific 
resistance  of  an  electrolyte — for  instance,  solutions  of  copper 
sulphate  of  different  concentrations.  The  box  form  of 
Wheatstone's  Bridge  is  most  suitable  for  the  purpose  (p.  1 54). 

A  steady  current  from  a  battery  and  a  galvanometer  to 
determine  when  there  is  a  balance,  as  ordinarily  used  with 
Wheatstone's  Bridge,  cannot  satisfactorily  be  used  in  meas- 
uring the  resistance  of  an  electrolyte,  for  a  steady  current 
produces  in  a  short  time  polarization  at  the  electrodes. 
This  polarization  leads  to  too  high  an  estimate  of  the  resist- 
ance of  the  electrolyte,  for  when  no  current  flows  through 
the  galvanometer,  the  three  other  arms  of  the  bridge  are 
balancing  the  potential  difference  necessary  to  overcome 
the  true  resistance  of  the  electrolyte  plus  the  potential 
difference  required  for  overcoming  the  polarization  potential 
difference  at  the  electrodes.  This  difficulty  is  obviated  by 
using  the  rapidly  alternating  current  from  the  secondary  of  an 
induction  coil  instead  of  a  steady  current  from  a  battery. 


SPECIFIC    RESISTANCE    OF    AN    ELECTROLYTE. 


189 


The  time  that  the  current  continues  in  one  direction  is  so 
short  that  no  appreciable  accumulation  can  form  at  the 
electrodes  to  produce  an  opposing  difference  of  potential. 
An  ordinary  galvanometer  would  not  be  affected  by  an 
alternating  current,  but  a  telephone  which  is  a  very  delicate 
detector  of  an  alternating  current  may  be  substituted. 

In  the  simplest  form  of  apparatus  a  vertical  glass  tube  of 
known  cross  section,  which  may 
be  found  by  calipers  (p.  14), 
holds  the  electrolyte.  The  elec- 
trodes are  connected  to  wires 
that  pass  through  the  stoppers; 
the  upper  electrode  can  be 
raised  or  lowered  as  desired. 
The  resistances  corresponding 
to  two  different  distances  of 
separation  of  the  electrodes 
should  be  determined.  From 
the  difference  we  get  the  re- 
sistance of  a  column  whose 
length  is  the  difference  in  the 
two  lengths  and  thus  eliminate  uncertainty  as  to  remaining 
polarization  at  the  electrodes  and  the  exact  ends  of  each 
column. 

Two  tubes  should  be  available  for  the  work.  In  one 
measurement  should  be  made  of  the  resistance  of  samples 
of  several  solutions  of  different  concentrations,  the  samples 
being  obtained  from  stock  bottles.  The  other  tube  is  for  the 
purpose  of  determining  the  temperature  coefficient  of  a 
solution.  It  is  placed  in  a  steam  heater  such  as  is  used  in 
Exp.  XIX  (p.  88).  As  the  heating  to  a  steady  temperature 
will  require  considerable  time,  the  tube  should  be  prepared 
at  the  beginning  of  the  work.  The  resistance  of  the  electro- 
lyte in  it  having  been  determined  at  room  temperature, 
the  heating  may  be  allowed  to  proceed  while  the  measure- 
ments with  the  other  tube  are  made. 

A  tube  of  small  cross  section  may  also  be  used.     If  its 


1 90  ELECTRICITY   AND    MAGNETISM. 

ends  pass  through  stoppers  into  much  larger  tubes  that 
contain  the  electrolyte  and  electrodes,  the  resistance 
measured  will  be  virtually  that  of  the  electrolyte  in  the 
small  tube.  The  diameter  of  the  tube  may  be  found  as  in 
Exp.  XI  (p.  58).  Substituting  a  different  length  of  the 
same  tubing  and  taking  differences  we  may  as  before 
eliminate  residual  polarization  effects. 

In  measuring  resistances  it  will  probably  be  desirable  to 
use  equal  resistances,  e.  g.,  100  ohms,  in  the  ratio  arms  of  the 
bridge.  It  may  be  impossible  to  obtain  a  balance  for 
which  there  is  no  sound,  for  even  though  there  were  a 
balance  for  steady  current,  there  would  not  in  general  be  a 
balance  for  varying  currents  such  as  are  used  in  this  experi- 
ment, owing  to  the  inductive  electromotive  forces  of  capacity 
and  self-induction  in  the  resistance  coils.  When  there  is 
uncertainty  as  to  whether  a  small  resistance  should  be  added 
or  cut  out,  the  ear  is  often  assisted  by  adding  and  cutting 
out  a  larger  resistance  about  which  there  is  no  doubt.  On 
comparing  the  change  of  tone  on  a  variation  of  this  latter 
resistance  with  the  variation  of  tone  with  the  uncertain 
resistance,  one  can  often  decide  whether  the  small  resistance 
should  be  added  or  not.  With  a  little  practice  one  should 
determine  resistances  within  i  per  cent. 

Calculate  the  specific  resistance  of  each  solution  at  each 
temperature  and  tabulate  the  results.  Find  also  the 
temperature  coefficient  of  the  solution  which  was  heated 
and  calculate  its  specific  resistance  at  o°. 

Questions. 

1 .  Why  should  we  expect  the  resistance  to  decrease  with  increased 
temperature  ? 

2.  What  is  supposed  to  be  the  nature  of  electric  conduction  in  an 
electrolyte? 

3.  Are  the  specific  resistances  inversely  proportional  to  the  con- 
centrations ?     Why  ? 


COMPARISON  OF  E.  M.  F/S  BY  HIGH-RESISTANCE  METHOD.     191 

LVI.  COMPARISON  OF  E.  M.  F.'S  BY  HIGH-RESISTANCE 

METHOD. 

Watson's  Practical  Physics,  pp.  430-431;  Stewart  and  Gee's  Practical 
Physics,  II,  pp.  101-102. 

The  readiest  method  of  comparing  the  electromotive  forces 
of  cells  is  by  means  of  a  galvanometer  of  sufficiently  high 
resistance.  If  the  deflections  are  (by  the  use  of  added 
resistances)  kept  small  the  deflections  of  the  galvanometer 
will  be  closely  proportional  to  the  currents  that  pass  through 
it  or  i  =  k.d  where  k  is  a  constant.  Two  methods  may  be 
employed  for  comparing  two  cells.  In  the  first,  called  the 
"equal  resistance"  method,  the  total  resistance  R  is  kept 
constant  (the  resistance  of  the  cells  being  supposed  negligi- 
ble). Hence,  by  Ohm's  Law,  the  e.  m.  f.'s  are  proportional 
to  the  currents,  that  is,  to  the  deflections,  or 


In  the  other  or  "equal  deflection  "  method,  such  resistances 
are  used  in  the  circuit  that  the  cells  cause  equal  deflections  of 
the  galvanometer.  Hence  by  Ohm's  Law,  since  the  currents 
are  equal,  the  electromotive  forces  must  be  proportional  to 
the  resistances,  or 

EjJZi 
E2     R, 

Both  methods  should  be  employed  to  find  the  e.  m.  f.'s 
of  several  cells  by  comparing  them  with  that  of  a  standard 
Daniell  cell  (p.  159).  Directions  for  the  adjustment  of  the 
telescope  and  scale  are  given  on  p.  25. 

(A)  Equal  Resistance  Method. — Make  R  such  that  the 
standard  Daniell  cell  gives  a  deflection  of  about  10  cm.  on 
a  scale  about  i  m.  from  the  mirror.  Make  a  reading  of  the 
zero;  i.  e.,  when  no  current  passes  through  the  galvanometer. 
Send  the  current  through  the  galvanometer  and  read  the 
division  now  on  the  cross-hair.  In  this  way  make  at  least 
six  readings  on  one  side  and,  reversing,  make  six  on  the 


I Q2  ELECTRICITY   AND    MAGNETISM. 

other.  Read  the  zero  often,  as  it  is  liable  to  change.  In 
reading,  use,  if  necessary,  the  method  of  vibration  (see  p. 
23).  If  the  vibrations  are  irregular  on  account  of  trolley 
currents  or  other  disturbances,  estimate  the  position  of  equi- 
librium from  the  vibrations  without  actually  making  read- 
ings. With  some  galvanometers  the  damping  is  so  great 
that  the  system  comes  to  rest  instead  of  vibrating  about  the 
position  of  equilibrium.  In  this  case  the  true  reading  can 
be  made  at  once.  Always,  if  possible,  estimate  tenths  of 
the  smallest  divisions.  When  you  have  thus  found  the 
mean  deflection  for  the  standard,  find  similarly  the  deflec- 
tion for  as  many  different  types  of  cells  as  time  allows. 
With  the  other  cells,  three  readings  on  a  side  will  be  suffi- 
cient. The  internal  resistance  of  the  different  batteries 
varies,  but  the  differences  are  negligible  compared  with  the 
total  resistance  of  the  circuit.  Express  in  volts  your  final 
values  of  the  e.  m.  f.'s  of  the  cells  tested. 

(B)  Equal  Deflection  Method. — With  a  resistance  which 
gives  a  deflection  of  about  10  cm.,  make  at  least  six  careful 
readings  of  the  deflection  on  each  side  given  by  the  standard 
Daniell  cell.  Replace  the  standard  by  one  of  the  cells 
to  be  tested  and  vary  the  resistance  of  the  circuit  until  the 
deflection  is  the  same  as  you  found  it  on  this  side  for  the 
standard.  Similarly  find  the  resistance  which  will  make 
the  deflection  on  the  other  side  the  same  as  that  given  by 
the  standard  on  that  side.  We  can  neglect,  in  comparison 
with  the  resistances  in  the  boxes,  the  resistance  of  the  bat- 
tery and  connecting  wires,  but  not  the  resistance  of  the  gal- 
vanometer. Take  the  mean  of  the  two  resistances  deter- 
mined above,  plus  the  resistance  of  the  galvanometer,  as  the 
resistance  required  to  give  the  same  deflection  as  the  stand- 
ard cell  gave  through  the  box-resistance  used  with  it,  plus 
the  galvanometer  resistance.  The  resistance  of  the  galvan- 
ometer, G,  must  be  determined  as  in  Exp.  XLV  (last 
paragraph) . 

In  determining  the  possible  error  of  your  results,  estimate 
the  possible  error  of  resistances  from  the  least  change  in 


COMPARISON  OF  E.  M.  F.'S  BY  CONDENSER  METHOD.      193 

resistance  which  will  have  an  appreciable  effect,  and  the 
possible  error  of  deflections  from  the  mean  deviation  from 
the  mean  in  your  readings. 

Questions. 

1.  What  are  the  advantages  and  disadvantages  of  the  type  of 
galvanometer  used  in  this  experiment  compared  with  other  types 
used  in  the  laboratory? 

2 .  Which  of  the  two  methods  do  you  consider  the  better  ?     Why  ? 

3.  How  could  this  method  be  used  for  finding  the  internal  resist- 
ance of  a  cell? 

4.  Are  the  deflections  of  a  galvanometer  strictly  proportional  to 
the  currents?     Why? 


LVII.  COMPARISON  OF  E.  M.  F.'S  AND  MEASUREMENT 

OF    BATTERY    RESISTANCE    BY    CONDENSER 

METHOD. 

Text-book  of  Physics  (Duff),  pp.  530-534,  562;  Watson's  Physics,  pp. 
634;  Watson's  Practical  Physics,  p.  526;  Henderson's  Electricity 
and  Magnetism,  pp.  185-187. 

When  a  condenser  of  capacity  C  is  connected  to  a  battery 
of  e.  m.  f.  E  it  receives  a  charge  Q  =  CE.  If  it  be  then  con- 
nected to  a  ballistic  galvanometer,  the 
throw,  d,  will  be  proportional  to  Q,  or  Q  = 
K  .  d,  where  K  is  a  constant.  We  shall  apply 
this  to  (A)  compare  e.  m.  f.  's  and  (B)  measure 
the  resistance  of  cells.  We  shall  describe 
these  separately,  but  in  practice  they  may 
be  combined. 

(A)  Suppose  the  condenser  is  first  charged 
by  a  battery  of  e.  m.  f.,  Ev  and  the  deflec- 

tion when  connected  to  the  ballistic  galva- 

, 
nometer  is  au  and  suppose  that  when  this 

same  condenser  has  been  charged  by  a  battery  of  e.  m.  f., 
E2,  the  deflection  is  d2;  then 

Kd,;  Q2=CE2=Kd2 
E      d  d 


FIG.  60. 


194 


ELECTRICITY   AND    MAGNETISM. 


Use  a  key  with  an  upper  and  a  lower  contact.  The 
condenser  should  be  connected  to  the  battery  when  the  key 
is  down  and  to  the  galvanometer  when  the  key  is  up.  Be 
very  careful  never  to  connect  the  battery  directly  to  the 
galvanometer.  When  a  discharge  is  sent  through  a  ballistic 
galvanometer,  the  needle  swings  over  to  one  side  and  then 
swings  back.  Observe  the  reading  of  the  scale  on  the  verti- 
cal cross-hair  of  the  telescope  when  the  needle  stops  and 
turns  back.  Always  in  such  work  estimate  tenths  of  the 
smallest  division.  Before  each  throw  bring  the  needle  as 
nearly  as  possible  to  rest.  The  zero  is  likely  to  change; 


|                         AUS 

P 

\\  | 

F 

|                            EIN 

FIG.  61. 

therefore,  before  each  throw,  record  the  zero,  and,  after  each 
throw,  record  both  the  turning-point  and  the  difference 
between  this  turning-point  and  the  zero;  i.  e.,  the  amount 
of  the  throw. 

Always  charge  the  condenser  for  approximately  the 
same  length  of  time,  for  instance,  five  seconds.  With  a 
standard  Daniell  cell  (p.  159),  record  six  throws  on  one  side. 
Reverse  the  battery  connections  and  record  six  throws  on 
the  other  side.  Let  the  mean  of  these  be  dr  Replace  the 
Daniell  cell  by  one  of  another  type  and  find  as  before  the 
mean  throw  d2.  If  E2  is  the  e.  m.  f.  of  the  latter  cell 


Measure  in  this  way  the  e.  m.  f.  of  as  many  cells  of  different 
type  as  possible. 

(B)   This  method  of  finding  the  resistance  of  a  cell  depends 
on  the  fact  that  when  the  poles  of  a  cell  of  resistance  B  are 


COMPARISON  OF  E.  M.  F.'s  BY  CONDENSER  METHOD.      195 

joined  by  a  conductor  of  resistance  r,  the  difference  of 
potential  of  the  poles  depends  on  the  ratio  of  r  to  B.  Let 
the  current  be  i.  Then  the  difference  of  potential  of  the 
poles  is  ri  by  Ohm's  Law  applied  to  the  part  r  of  the  circuit. 
If  in  this  condition  the  poles  be  joined  to  a  condenser  of 
capacity  C  it  will  receive  a  charge  Cri  and  if  this  when 
discharged  through  a  condenser  causes  a  throw  df 

Cri=K-d' 

Now  if  E  be  the  e.  m.  f.  of  the  cell,  E=(B  +r)i  by  Ohm's 
Law  applied  to  the  whole  circuit.  Hence,  if  the  condenser  be 
charged  by  the  cell  when  it  is  not  short-circuited  as  above, 
the  charge  C  E  also  equals  Ci(B  +r)  and  if  the  deflection 
when  the  condenser  is  discharged  is  d, 


Dividing  one  equation  by  the  other  and  solving  for  B, 

d-d' 
B=r~^-  •.. 

The  connections  are  the  same  as  when  comparing  e.  m.  f.'s 
with  the  addition  of  a  circuit  containing  a  resistance 
and  a  very  low  resistance  key  (e.  g.,  a  mercury  key),  con- 
necting the  poles  of  the  cell.  The  battery  should  be  short- 
circuited  just  before  the  charging  key  is  depressed,  and  the 
short-circuiting  key  should  be  released  immediately  after 
the  other,  otherwise  the  battery  will  run  down.  Choose 
such  a  short-circuiting  resistance  that  the  galvanometer 
throw  is  reduced  to  about  half  the  value  which  it  has  without 
the  short-circuit.  Do  not  use  the  plug-box  resistances 
for  this  work,  on  account  of  the  danger  of  burning  them  out, 
but  use  open  wound  resistances  of  large  wire..  Find  the 
internal  resistance  of  cells  of  several  different  types. 

In  estimating  the  possible  error  of  your  results,  estimate 
the  possible  error  of  your  mean  readings  from  the  mean 
deviation  from  the  mean  in  the  individual  readings. 

Additional  exercises  (to  be  performed  if  time  permit.) 

(C)   Study  the  effect  of  length  of  time  of  charge  by  means 


196  ELECTRICITY   AND    MAGNETISM. 

of   the   throws   obtained   with  the   condenser   charged   for  • 
different  lengths  of  time  with  the  same  battery. 

(D)  Study  the  leakage  of  the  condenser  by  comparing  the 
throws  when  the  condenser  has  been  successively  charged 
with  the  same  e.  m.  f.,  and  has  remained  charged  for  different 
intervals  of  time. 

(E)  Study   the   electric   absorption   of   the    condenser   by 
charging  for  several  minutes,  discharging  and  reading  the 
throw  and  immediately  insulating;  after  one  minute,  again 
discharge   and  insulate.     Continue   discharging  for   several 
minutes,  the  condenser  being  insulated  during  the  minute 
intervals. 

Questions. 

1.  What  are  the  peculiarities  and  requirements  of  a  good  ballistic 
galvanometer  ? 

2.  What  is  the  construction  of  a  condenser  and  what  do  absorption 
and  leakage  mean? 

3.  How  could  you  find  the  resistance  of  the  galvanometer  used,  em- 
ploying a  condenser  and  a  known  resistance  ? 

LVIII.  CALIBRATION    OF    A    VOLTMETER. 

Ames1  General  Physics,  pp.  674-675;  Text-book  of  Physics  (Duff), 
PP-  57°~57 1;  Watson's  Practical  Physics,  pp.  493-498;  Hender- 
son's Electricity  and  Magnetism,  pp.  200-208;  Hadley's  Electricity 
and  Magnetism,  p.  320. 

A  voltmeter  may  be  calibrated  by  balancing  a  part  of  the 
e.  m.  f.  applied  to  the  terminals  of  the  voltmeter  against 
the  e.  m.  f.  of  one  or  more  standard  cells.  To  do  this  a 
very  high  resistance  circuit,  consisting  of -a  high  resistance- 
box  in  series  with  an  ordinary  resistance-box,  is  placed  in 
parallel  with  the  voltmeter.  The  fall  of  potential  in  part 
of  the  low  resistance-box  is  measured  by  a  side-circuit  con- 
sisting of  the  standard  cell,  a  sensitive  galvanometer  and  a 
key,  the  standard  cell  being  so  turned  that  it  tends  to 
send  a  current  through  the  galvanometer  in  the  opposite 
direction  to  the  fall  of  potential  in  the  box. 

Let  e  be  the  e.  m.  f.  of  the  cell,  E  the  potential  difference 
at  the  terminals  of  the  voltmeter,  r1  the  resistance  across 


CALIBRATION    OF    A    VOLTMETER. 


197 


which  the  galvanometer  circuit  is  connected,  and  r^  the 
remaining  resistance  in  the  high-resistance  circuit.  When 
rl  is  adjusted  so  that  there  is  no  deflection  when  the  key  is 
pressed 


A  special  fuse-wire  for  very  low  currents  should  be  placed 
immediately  adjacent  to  the  battery  to  prevent  the  possi- 
bility of  injury  to  the  resistance-boxes.  The  main  circuit 
should  be  closed  through  a  spring-key  only  a  sufficient 
length  of  time  to  enable  the  voltmeter  or  the  galvanometer 
to  be  read.  A  high  resistance  should  be  placed  in  series 
with  the  standard  cell  to  prevent  any  considerable  current 
passing  through  it.  In  first 
performing  this  experiment 
it  is  well  to  use  a  simple 
and  inexpensive  f6rm  of 
standard  cell,  and  the 
Daniell  cell  (p.  159)  will  be 
suitable.  For  later  and 


more  accurate  work,  either 
the  Clark  or  the  Weston  cell 
should  be  used.  By  vary- 
ing the  number  of  cells  in 
the  main  circuit  or  using  different  resistances  in  the  main 
circuit,  different  voltages  at  the  terminals  of  the  voltmeter 
may  be  obtained. 

The  above  method  will  not  apply  if  the  voltmeter  is  to 
be  tested  at  voltages  less  than  the  e.  m.  f.  of  the  standard 
cell.  In  this  case,  an  inversion  of  the  connections  may  be 
used.  Instead  of  balancing  a  variable  part  of  the  voltage 
against  the  e.  m.  f.  of  the  cell  a  variable  part  of  the  e.  m.  f. 
of  the  cell  is  balanced  against  the  voltage.  A  little  con- 
sideration will  indicate  the  necessary  change  of  connection. 

Instead  of  the  above  temporary  arrangement  of  circuits, 
a  Potentiometer,  which  consists  essentially  of  the  several 


198  ELECTRICITY  AND    MAGNETISM. 

circuits  with  the  necessary  resistances  and  keys  in  perma- 
nent connection,  may  be  used.  With  its  aid  the  work  may 
be  performed  more  rapidly  and  more  accurately.  Its  parts 
and  connections  should  be  carefully  traced  out  with  the 
assistance  of  a  large  diagram  (which  may  be  attached  to  the 
wall  near  the  instrument)  and  additional  explanation  will  be 
supplied  by  the  instructor. 

A    calibration    curve,    consisting    of    true    volts    plotted 
against  scale  readings,  should  be  drawn. 

Questions. 

1.  Prove  the  above  formula  by  applying  Kirchoff's  laws. 

2.  Draw  a  diagram  showing  the  connections  when  a  millivoltmeter 
has  to  be  calibrated. 

3 .  Draw  a  diagram  to  show  how  the  above  method  could  be  adapted 
to  compare  the  e.  m.  f.'s  of  cells. 


LIX.  CALIBRATION    OF   AMMETER. 

Hadley's  Electricity  and  Magnetism,  p.  325;  Watson's  Practical 
Physics,  pp.  516—517;  Henderson's  Electricity  and  Magnetism,  p. 
205. 

A  method  somewhat  similar  to  that  used  for  the  volt- 
meter may  be  employed.  The  current  from  a  storage 
battery  that  passes  through  the  ammeter  passes  also  through 
a  conductor  of  large  current  capacity  and  of  measured 
resistance  and  a  switch.  The  potential  difference  at  the  ends 
of  this  conductor  is  found  by  a  shunt  circuit,  consisting  of 
a  high  resistance-box  in  series  with  a  box  containing  low 
resistances.  In  parallel  with  the  latter  is  a  circuit  containing 
a  Daniell  cell,  a  sensitive  galvanometer  and  a  key.  A 
special  very  fine  fuse-wire  should  be  used  in  series  with  the 
two  boxes,  and  its  resistance  should  be  known  and  taken 
account  of  in  calculating  the  current.  The  large  conductor 
should  be  immersed  in  oil  and  its  temperature  kept  as  nearly 
as  possible  at  the  temperature  at  which  its  resistance  is 
determined.  To  prevent  heating,  the  main  current  should 
be  closed  for  short  intervals  only. 


CALIBRATION    OF   AMMETER. 


199 


The  galvanometer  should  be  protected  by  a  shunt  during 
the  first  adjustments.  Notice  first  in  which  direction  the 
galvanometer  moves  when  the  key  in  its  circuit  is  depressed. 
The  deflection  should  be  reversed  or  reduced  when  in 
addition  the  switch  is  closed.  If  this  is  found  not  to  be  so, 
the  connection  of  either  the  Daniell  cell  or  the  storage 
batteries  should  be  reversed.  The  resistance  in  the  box 
nearest  the  galvanometer  and,  if  necessary,  in  the  other 
box  also,  should  be  varied 
until  there  is  no  deflection  if 
the  galvanometer  key  is  de- 
pressed when  the  switch  is 
closed. 

When  the  adjustment  has 
been  obtained  as  closely  as 
possible,  the  fall  of  potential 
between  the  points  of  the 
high-resistance  circuit  to 
which  the  standard  cell  circuit 
is  attached  equals  the  e.  m.  f. 
of  the  cell  (p.  159).  From  this 
and  the  resistances  of  the  boxes  the  fall  of  potential  between 
the  ends  of  the  large  conductor  is  found  and  then  from  the 
resistance  of  the  large  conductor  the  current  through  it 
and  the  ammeter  is  calculated.  The  total  resistance  in 
the  two  boxes  must  be  kept  high.  A  preliminary  calculation 
will  show  about  how  large  the  resistance  of  large  capacity 
should  be.  A  number  of  currents  distributed  over  the 
range  of  the  ammeter  should  be  used  and  from  the  results  a 
calibration  curve  should  be  drawn. 

In  the  above  we  have  assumed  that  the  voltage  applied 
to  the  conductor  exceeds  that  of  the  cell.  If  the  reverse  is 
the  case,  the  arrangement  must  be  inverted;  i.  e.,  part  of 
the  e.  m.  f.  of  the  cell  must  be  balanced  against  the  voltage 
applied  to  the  conductor.  This  method  must  be  applied 
when  the  current  is  less  than  the  quotient  of  the  e.  m.  f.  of 
the  cell  and  the  resistance  of  the  conductor. 


200  ELECTRICITY   AND    MAGNETISM. 

Instead  of  the  arrangements  of  circuits  above,  the  Po- 
tentiometer referred  to  in  Exp.  LVIII  may  be  used  to 
measure  the  fall  of  potential  in  the  conductor  of  large  cur- 
rent capacity. 

Questions. 

1.  Why  must  the  resistance  in  the  shunt  circuit  be  large? 

2.  Storage  batteries  giving  an  e.  m.  f.  of  50  volts  are  available  for 
calibrating  an  ammeter  whose  range  is  from  5  to  25  amperes  and 
whose  resistance  is  0.2  ohms,      (a)  What  is  the  least  possible  value 
for   the   resistance    of    large    capacity?      (b)   What    is    the    greatest 
value? 


LX.  COMPARISON    OF    CAPACITIES    OF    CONDENSERS. 

Henderson1 s 
's    Practical 


Hadley's    Electricity    and    Magnetism,    pp.     331—334;     Henderson's 
Electricity    and    Magnetism,    pp.    235—241;    Watson' j 


Physics,  pp.  530-535. 

Two  or  more  condensers  are  to  be  compared  by  three 
methods. 

(A)  First  Method.  —  Each  condenser  is  charged  in  turn 
by  the  same  battery  and  then  discharged  through  a  ballistic 
galvanometer.  Let  the  capacities  of  the  two  condensers  be 
C\  and  C2.  The  charges  which  they  receive  when  connected 
to  a  battery  of  e.  m.  f.  E,  are  Q1=C1E,  and  Q2.=C2E.  Let 
the  throws  of  the  galvanometer  when  the  condensers  are 
discharged  through  it  be  dl  and  d2,  respectively.  Then 


(Exp.  LVII).  The  connections  are  the  same  as  in  Exp.  LVII, 
with  the  addition  of  one  or  more  keys  to  charge  alter- 
nately two  or  more  condensers.  If  either  deflection  be  too 
small,  additional  cells  should  be  added.  Storage  cells  may 
be  used  if  connections  are  made  through  special  very  fine 
fuse-wires  to  protect  the  resistances. 

If  either  deflection  be  too  large  the  galvanometer  should  be  shunted 
by  a  known  resistance,  S.  Let  G  be  the  resistance  of  the  galvan- 
ometer determined  as  in  Exp.  XLV  (last  paragraph),  d'  the 
throw  obtained  with  the  galvanometer  shunted,  d  the  throw  which 


COMPARISON    OF    CAPACITIES    OF    CONDENSERS. 


201 


would  have  been  obtained  without  the  shunt,  qr  the  quantity  of 
electricity  passing  through  the  galvanometer,  q"  the  quantity  passing 
through  the  shunt.  Then 

9"_5 
q'     S 

for  charges  of  electricity,  like  steady  direct  currents,  divide  inversely 
as  the  resistances  (p.  158).  Hence 


Since 


£_* 

q'     d" 


(B)  Bridge  Method.  —  The  two  condensers  to  be  com- 
pared, C\  and  C2,  form  two  arms  of  a  Wheatstone's  Bridge, 
two  high  non-inductive 
resistances,  R1  and  R2  (see 
figure),  preferably  several 
thousand  ohms,  forming 
the  other  two  arms.  These 
two  resistances  are  ad- 
justed until  on  closing  the 
battery  circuit  at  a  the 
galvanometer  is  not  dis- 
turbed. Then  during  both  r~^ 
charge  and  discharge  the 
farther  poles  of  the  con- 
denser (A  and  B)  must 
remain  at  the  same  potential  as  well  as  the  nearer  poles 
(joined  at  D).  Hence  the  charges  Q^  and  Q2  in  the  con- 
densers must  have  the  ratio,  Qt  :  Q2'  :  C\  :  C2.  But  the  quanti- 
ties which  have  flowed  into  the  condensers  will  be  inversely 
proportional  to  the  resistances  through  which  the  charges 
have  flowed,  that  is,  Q^\  Q2:  :  R2:  R^  Hence 


FIG.  65. 


C2     R,' 

The  battery  key  has  an  upper  and  lower  contact;  the 
upper  contact  (b),  against  which  the  lever  ordinarily  rests, 


2O2 


ELECTRICITY   AND    MAGNETISM. 


short-circuits  the  battery  terminals  of  the  bridge,  thus  keep- 
ing the  condenser  uncharged.  The  sensitiveness  may  be 
increased  by  increasing  the  number  of  cells  in  the  battery, 
and  also  by  using  a  double  commutator  (see  p.  1 6 1  j .  Instead 
of  a  galvanometer  a  telephone  may  be  used  in  this  method, 
the  battery  being  replaced  by  a  small  induction  coil. 

(C)  Thomson's  Method  of  Mixtures. — The  connections 
are  as  shown  in  the  figure.  Kl  is  a  Pohl's  commutator,  K2 
an  ordinary  single  contact  switch.  When  the  swinging 


arm  of  the  commutator  is  in  the  position  aa',  the  two  con- 
densers are  charged,  Cl  to  the  difference  of  potential  at  the 
extremities  of  Rlf  C2  to  the  difference  of  potential  at  the 
extremities  of  R2.  The  swinging  arm  of  the  commutator  is 
now  placed  in  the  position  b'b'  and  the  two  charges  are 
allowed  to  mix.  If  they  are  exactly  equal,  being  of  opposite 
sign,  the  galvanometer  will  not  be  affected  when  K2  is 
depressed.  Rl  and  R2  (which  are  large  resistances,  pref- 
erably several  thousand  ohms),  are  adjusted  until  this  is 
secured.  The  charges  being  equal,  C1Vl  =  C2V2,  and  since 


ABSOLUTE    MEASUREMENT    OF    CAPACITY.  203 

Questions. 

1.  What  is  the  composite  capacity  of  three  microfarad  condensers 
in  parallel?     In  series? 

2.  Which  of  these  three  methods  do  you  consider  best? 

3.  State  briefly  in  words  (without  formulae)   why  charges  divide 
like  steady  currents;  i.  e.,  inversely  as  the  ohmic  resistances. 

4.  Why   cannot   series  resistance  be  used  in    (A)    to   reduce  the 
sensitiveness  of  the  galvanometer. 


LXI.  ABSOLUTE  MEASUREMENT  OF  CAPACITY. 

The  magnitude  of  a  capacity  can  also  be  found  without 
the  use  of  a  known  capacity  with  which  to  compare  it.  This 
can  be  done  in  different  ways.  The  following  is  one  of  the 
simplest. 

Let  Q  be  the  charge  received  by  the  condenser  of  unknown 
capacity  C  when  connected  to  a  cell  of  known  e.  m.  f.,  E. 
Then 


To  find  Q  discharge  the  condenser  through  a  ballistic  galvan- 
ometer the  constant  of  which  has  been  found  by  the  method 
of  Exp.  LXIV.  If  the  constant  be  K  and  the  deflection 
Dt  Q  =  KD. 


LXII.  COEFFICIENTS  OF  SELF-INDUCTION  AND  OF 
MUTUAL  INDUCTION. 

Text-book  of  Physics  (Duff),  pp.  609-611;  Ames'  General  Physics, 
PP-  743-745;  Watson's  Practical  Physics,  pp.  543-548;  Parr's 
Practical  Electrical  Testing,  p.  207;  Hadley's  Electricity  and 
Magnetism,  pp.  417-422. 

The  coefficient  of  self-induction  of  a  circuit  is  the  number 
of  magnetic  lines  of  force  which  link  with  the  current  when 
the  circuit  is  traversed  by  unit  current.  Owing  to  the 
difficulty  of  calculating  this  important  quantity  from  the 
dimensions  of  the  circuit,  experimental  methods  of  deter- 
mination have  much  value. 


204 


ELECTRICITY   AND    MAGNETISM. 


(A)  Probably  the  best  method  (using  direct  currents) 
is  Anderson's  Modification  of  Maxwell's  Method.  The 
connections  are  shown  in  the  figure.  The  coil  of  self-in- 
duction L  and  resistance  Q  is  made  one  arm  of  a  Wheat- 
stone's  Bridge  (preferably  Post-office  Box  form).  Ob- 
tain a  balance  for  steady  currents  by  proper 
variation  of  5,  so  that  when  Kl  is  closed 
and  then  K2,  the  galvanometer  is  not  dis- 
turbed. For  delicacy  of  adjustment  it  is 
well  to  either  have  a  resistance  which  can 
be  varied  continuously  form  a  part  of  5, 
or  make  the  ratio  arms  P  and  R  such  that 
5"  is  large.  Vary  r,  the  resistance  in  the 
battery  circuit,  and  if  necessary,  vary  the 
capacity  of  the  condenser  C  until  there  is 
a  balance  for  transient  currents;  i.  e., 
until  the  galvanometer  is  not  disturbed 


FIG.  67. 


when  K2  is  depressed  and  Kl  depressed  afterward.     Then 
if  C  is  the  capacity  of  the  condenser. 


For  at  time  t\et  x  =  current  in  branch  AB,  y  =  current  in  AD  =  cur- 
rent in  D  E,  0  =  current  in  BE  .'.  current  in  r  =  y  +  z.  q  =  charge  in 
condenser,  e  =  potential  difference  of  its  poles  =  Rz  +  r  (y  +  z)  . 

Since  there  is  a  balance  for  transient  currents  we  may  equate  the 
e.  m.  f.  in  AD  to  that  in  AB.  Hence 


The  current  in  the  branch  containing  the  condenser  is  (x  —  z)  ;  but 
it  can  also  be  expressed  as 


dq       rde 
dJmCdi 


Hence 


Now  since  there  is  a  balance  for  steady  currents  RQ  =  PS  and 
since  Rz  =  Sy,  it  readily  follows  that 


=  Qy  +  C[r 


COEFFICIENTS    OF    INDUCTION. 

If  the  resistances  are  expressed  in  ohms  and  the  capac- 
ity in  farads,  the  results  will  be  in  henries. 

Measure  the  self-induction  of  a  coil  whose  length  is 
great  compared  with  its  diameter  and  compare  the  result 
with  that  calculated.  To  calculate  the  coefficient  of  self- 
induction  it  is  necessary  to  know  the  number  of  lines  of 
force  passing  through  the  coil.  This  number  multiplied  by 
the  number  of  turns  will  give  the  number  which  link  with 
the  current;  i.  e.,  the  self-induction.  If  A  be  the  area  of 
the  cross  section  of  a  solenoid  of  practically  infinite  length, 
with  nQ  turns  per  cm.  of 
length,  the  number  of  lines 
is  4xn0A  for  unit  current. 
The  number  of  turns  in  a 
length  d  is  nQd;  hence  the 
coefficient  of  self-induction  of 
this  length  is  47iAnQ2d  in  C. 
G.  S.  units.  Reduce  to 
henries  by  dividing  by  io9 
(p.  162).  ' 

To  secure  greater  sensitive- 
ness in  making  the  balance  FlG 
for  transient  currents,  replace 

the  battery  by  the  secondary  of  a  small  induction  coil  and 
the  galvanometer  by  a  telephone,  or  use  a  double  commu- 
tator (see  p.  161). 

(B)  Comparison  of  Two  Coefficients  of  Self-induction. — 
The  two  coils  of  self-induction  Llf  L2,  and  resistances  Rv 
R2,  are  placed  in  two  arms  of  a  Wheatstone's  Bridge,  a 
variable  resistance,  r,  being  included  in  one  arm.  By  vary- 
ing r,  and,  if  possible,  by  varying  one  of  the  self-inductances, 
if  not,  by  varying  r,  P,  and  Q,  find  a  balance  for  both  steady 
and  transient  currents. 

Then  for  steady  currents 


206  ELECTRICITY   AND    MAGNETISM. 

and  for  transient  currents 

P 


where  <o  =  271 X  frequency.     Hence 


The  above  adjustment  is  obtained  by  first  securing  a 
balance  for  steady  currents.  A  balance  for  transient  cur- 
rents is  then  sought  by  varying  Ll  or  L2.  If  this  cannot 
be  secured,  r  and  P  must  be  both  increased  or  decreased 
and  a  balance  for  steady  currents  again  obtained  and  then 

one  for  transient  currents  found  by 
varying  Lt  or  L2.  If  neither  Ll 
nor  L2  can  be  varied,  a  balance  can 
only  be  obtained  by  a  series  of  trials 
as  above,  the  ratio  of  P  and  (7^  +r) 
being  kept  constant,  so  that  the 
steady  current  balance  may  not  be 
disturbed.  To  increase  the  sensi- 
tiveness with  transient  currents,  a 
double  commutator  (p.  161)  should 
be  used  or  the  battery  may  be  re- 
placed by  the  secondary  of  an  induction  coil  and  the  galva- 
nometer by  a  telephone. 

(C)  The  coefficient  of  mutual  induction  of  two  coils  is  the 
number  of  lines  of  force  which  link  with  the  turns  of  the  other 
when  the  first  is  traversed  by  unit  current.  Pirani's 
method  is  perhaps  the  most  satisfactory  for  the  experimental 
determination  of  coefficients  of  mutual  induction.  The 
connections  are  shown  in  figure  69.  If  M  be  the  required 
coefficient,  C  the  capacity  of  the  condenser,  and  r^  and  r2 
the  values  of  the  variable  resistances  for  which  the  galvan- 
ometer is  not  disturbed, 


FIG.  69. 


STRENGTH    OF   A    MAGNETIC    FIELD.  207 

For  let  the  steady  current  in  the  battery  circuit  =  i.  The  potential 
difference  at  the  terminals  of  rlt  =  ir^  The  charge  of  the  condenser  is 
Cirr  If  t  —  time  required  to  establish  or  destroy  the  battery  current 
the  average  current  in  the  condenser  branch  during  this  time  = 
Cirl/t  and  the  potential  difference  at  the  terminals  of  r.2  =  Cirlr.J/t. 
Opposing  this  e.  m.  f.  in  the  galvanometer  circuit  is  that  due  to  M, 
the  average  value  of  which  =  Mi/t.  If  the  galvanometer  is  not  dis- 
turbed on  making  or  breaking  the  battery  circuit,  Cir^r^t^Mi/t 
.'.  M  =  Crlr.2. 

The  secondary  of  M  is  acted  on  by  two  e.  m.  f.'s — one  due 
to  its  connection  with  the  main  circuit  in  which  there  is  an 
e.  m.  f.,  the  other  due  to  mutual  induction  between  the 
primary  and  secondary.  If  these  two  do  not  oppose  one 
another  a  balance  cannot  be  found.  If  such  is  found 
to  be  the  case  the  connections  of  the  secondary  to  the 
galvanometer  circuit  must  be  reversed.  To  increase  the 
sensitiveness  of  the  method  a  double  commutator  may  be 
used  or,  better  still,  the  battery  and  galvanometer  may  be 
replaced  by  a  small  induction  coil  and  telephone. 

When  an  approximate  adjustment  has  been  found  C  and 
rl  should  be  altered  until  the  sensitiveness  is  a  maximum, 
and  then  rl  and  r2  treated  in  the  same  way. 

Find  the  coefficient  of  mutual  induction  of  two  coils,  one 
wound  upon  the  other,  and  one  of  which  is  long  compared 
with  its  diameter,  and  compare  the  result  with  that  calculated 
from  the  definition. 

QUESTIONS. 

1.  Coils  with  iron  cores  do  not  have  definite  induction  coefficients. 
Explain. 

2.  How  are  resistance  coils  in  boxes  wound  so  as  to  be  free  from 
self-induction  ? 


LXIII.  STRENGTH  OF  A  MAGNETIC  FIELD  BY  A 
BISMUTH  SPIRAL. 

Ames'  General  Physics,  p.   758;  Hadley' s  Electricity  and  Magnetism, 

p.  296. 

The  electrical  resistance  of  a  bismuth  wire  is  changed 
when  it  is  placed  transverse  to  a  magnetic  field  and  the 
magnitude  of  the  change  depends  on  the  strength  of  the  field. 


208  ELECTRICITY    AND    MAGNETISM. 

When  a  curve  representing  the  resistance  of  a  flat  spiral 
of  bismuth  as  a  function  of  the  strength  of  the  magnetic 
field  has  been  obtained  the  spiral  may,  in  connection  with  a 
Wheatstone's  Bridge,  be  used  to  measure  the  strength  of  any 
magnetic  field  within  the  range  of  the  calibration.  For 
instance,  it  may  be  used  to  study  the  magnetic  field  of  an 
electromagnet.  The  following  three  points  may  be  examined : 

(A)  Find  how  the  magnetic  field  between  the  poles  varies 
when   the   strength   of  the  current  actuating  the  electro- 
magnetic is  varied  by  means  of  a  rheostat. 

(B)  Find  how  the  strength  of  the  field  midway  between 
the  pole-pieces  changes  when  the  distance  apart  of  the  pole- 
pieces  is  varied,  the  current  being  kept  constant. 

(C)  Find  how  the  strength  of  the  field  in  an  equatorial 
plane  varies  with  the  distance  from  the  axis  of  the  pole-pieces. 

In  each  case  represent  the  results  by  means  of  a  curve. 


LXIV.  CONSTANT  AND   RESISTANCE   OF   A   BALLISTIC 
GALVANOMETER. 

Text-book  of  Physics  (Duff),  pp.  556,  562;  Ames'  General  Physics,  p. 
674,  712;  Watson's  Practical  Physics,  pp.  518-524;  Pierce,  Am. 
Acad.  Arts  and  Sc.  Vol.  42,  pp.  159—160. 

When  quantities  of  electricity  are  discharged  through  a 
ballistic  galvanometer  (p.  156)  the  throws  are  proportional 
to  the  quantities  or, 

Q=K.d, 

where  K  is  the  constant  of  the  ballistic  galvanometer.  To 
determine  the  value  of  K  a  known  quantity  must  be  dis- 
charged through  the  galvanometer  and  the  throw  noted. 

This  known  quantity  might  be  obtained  from  a  condenser 
of  known  capacity,  charged  to  a  known  potential,  or  by 
turning  an  earth  inductor  (Exp.  XLIII)  in  a  field  of 
known  strength.  Both  of  these  methods,  however,  re- 
quire that  other  constants  (capacity  and  e.  m.  f.  in  the  first, 
strength  of  field  in  the  second)  be  determined. 


BALLISTIC    GALVANOMETER.  2OQ 

A  simpler  method  is  to  use  a  so-called  calibra  ting-coil  ; 
i.  e.,  an  induction  coil  of  known  winding  without  a  magnetic 
core.  The  primary  is  a  long,  straight  helix,  so  long  that 
there  is  no  appreciable  leakage  near  the  center.  Over  the 
center  there  is  wound  a  secondary.  If  the  primary  be  of  n 
turns  per  cm.  and  the  secondary  be  of  n'  total  turns,  then 
the  magnetizing  force  produced  by  a  current  of  i  amperes 
in  the  primary  is 


10 

and  the  quantity  induced  in  the  secondary  by  making  or 
breaking  i  is 

han' 


where  a  is  the  area  of  cross  section  of  the  cylinder  on  which 
the  primary  is  wound,  r  is  the  total  resistance  of  the  second- 
ary circuit  and  the  factor  io8  is  required  when  q  is  in 
coulombs  and  r  in  ohms  (p.  162). 

The  number  of  lines  of  force  that  pass  through  the  secondary  is 
a  h.  Hence  when  h  is  increasing  the  induced  e.  m.  f.  is  in  absolute 
units 

,d(ah) 

The  quantity  induced  is     I   i  dt  and  i  equals  e/r.     Hence 

Jn'  d(ah)     n'ak 
7~dT  '—  • 


From  the  above  expression  for  q  and  the  throw  d,  K  can 
be  calculated.  If  the  throw  is  small  it  may  be  doubled  by 
reversing  i  and  the  half  of  the  double  throw  taken  for  d. 
Several  currents  should  be  tried.  The  value  of  K  thus 
found  is  in  coulombs  per  scale  division.  If  the  distance  of 
the  scale  from  the  galvanometer  be  changed  in  any  pro- 
portion K  will  be  increased  in  the  same  proportion.  Hence 
the  distance  should  be  noted  unless  it  is  not  liable  to  change. 

It  is  often  necessary  to  change  the  sensitiveness  of  a  ballistic 
galvanometer  by  shunting  it  or  putting  resistance  in  series 
14 


210  ELECTRICITY    AND    MAGNETISM. 

with  it.  To  allow  for  this  we  must  know  the  resistance  of 
the  galvanometer. 

The  resistance  of  a  ballistic  galvanometer  when  used 
ballistically  on  a  closed  circuit  is  different  from  its  resistance 
when  used  as  an  ordinary  galvanometer.  This  is  due  to  the 
fact  that  the  galvanometer  coil  moves  in  a  magnetic  field, 
and  thus  an  induced  e.  m.  f.  is  produced,  which  opposes  the 
applied  e.  m.  f.  and  has  the  same  effect  as  an  added  resist- 
ance. To  find  the  effective  resistance  of  a  ballistic  gal- 
vanometer we  may  use  the  same  apparatus  and  connections 
as  in  finding  the  constant  of  the  galvanometer.  If  a 
current  be  reversed  in  the  primary  of  the  calibrating  coil, 
the  quantity  of  electricity  that  will  flow  through  the  second- 
ary will  vary  inversely  as  the  total  secondary  resistance. 
Hence,  by  observing  the  throw  with  a  certain  primary 
current,  and  then  increasing  the  secondary  resistance  by  the 
insertion  of  a  box-resistance  and  repeating  the  reversal  of 
the  primary,  we  can,  by  proportion,  find  the  resistance  of 
the  galvanometer  when  used  ballistically. 

The  resistance  of  the  galvanometer  should  also  be  found 
by  the  method  of  Exp.  XLVI  or  that  of  Exp.  XLV  and 
compared  with  the  above. 

LXV.  MAGNETIC  PERMEABILITY. 

Text-book  of  Physics  (Duff),  pp.  500—503;  Ames'  General  Physics,  pp. 
609,  615;  Watson's  Physics,  pp.  712-714;  Hartley's  Electricity 
and  Magnetism,  pp.  384-392;  Watson's  Practical  Physics,  pp. 
563-568;  Henderson's  Electricity  and  Magnetism,  pp.  282-284; 
Swing's  Magnetism  in  Iron,  Chapter  III. 

A  current  in  a  long  solenoid  of  wire  will  produce  near 
the  center  of  the  solenoid  a  magnetic  force  H,  which  may 
be  specified  by  the  number  of  lines  of  force  per  unit  of  area 
at  right  angles  to  the  lines.  If  a  long  iron  rod  be  now 
thrust  into  the  solenoid,  the  number  of  lines  of  force  (now 
called  lines  of  induction)  will  be  much  greater,  say  B  per 
unit  of  area.  The  permeability  of  the  iron  is  defined  as 
u=B+H. 


MAGNETIC    PERMEABILITY.  211 

If  this  experiment  were  performed  with  comparatively 
short  iron  rods,  it  would  be  found  that  B  would  be  less  the 
shorter  the  rod.  One  consistent  way  of  explaining  this  is 
to  consider  the  free  poles  developed  at  the  ends  of  the  rod 
when  magnetized.  A  little  consideration  will  show  that 
they  of  themselves  would  produce  a  magnetic  force  in  the 
space  occupied  by  the  iron,  this  magnetic  force  being  opposed 
to  the  original  magnetizing  force,  and  so  we  may  say  that 
the  effective  magnetic  force,  H,  is  the  original  magnetic 
force  diminished  by  the  demagnetizing  force  of  the  poles. 
It  is  this  effective  magnetic  force  that  we  should  divide  into 
the  induction  to  get  the  permeability.  The  calculation  of 
the  demagnetizing  force  is  usually  difficult  and  uncertain, 
and  so  it  is  better  to  take  some  method  of  eliminating  it. 

One  such  way  is  that  implied  in  the  statement  at  the  out- 
set, to  use  a  long  rod,  for  that  will  diminish  the  magnitude 
of  the  demagnetizing  force  at  the  center.  But  the  necessary 
length  makes  it  inconvenient  to  test  specimens  in  this  way. 
Another  method  is  to  join  the  ends  of  the  rod  by  a  heavy 
yoke  of  iron,  for  opposite  poles  developed  in  the  yoke  neu- 
tralize the  effect  of  the  poles  in  the  rod.  (This  is  one  way  of 
stating  the  case.  Another  way  is  to  say  that  the  yoke  carries 
around  the  lines  of  force.  A  third  way  is  to  say  that  the 
yoke  diminishes  the  magnetic  resistance  of  the  circuit.) 
The  difficulty  with  a  yolk  method  is  in  getting  a  satisfactory 
contact  between  yoke  and  rod.  A  very  small  gap  will  re- 
sult in  the  neutralization  being  not  quite  complete  (or  in 
leakage  of  lines  of  force  or  in  magnetic  resistance)  . 

A  more  satisfactory  method  is  to  take  an  endless  speci- 
men; i.  e.,  a  ring.  Then  there  are  no  free  poles  and  no  de- 
magnetizing force.  On  the  ring  a  magnetizing  coil  of  N 
turns  per  cm.  is  wound.  When  a  current  of  /  amperes  passes 
through  it,  the  magnetizing  force  produced  is 


IO 

For  finding  the  value  of.  B  a,  secondary  coil  is  wound  on  the 


212 


ELECTRICITY   AND    MAGNETISM. 


ring  and  put  in  series  with  a  ballistic  galvanometer.  Sup- 
pose the  iron  initially  free  from  magnetism.  The  setting  up 
of  the  field  B  produces  a  discharge,  Q,  of  electricity  through 
the  secondary.  If  A  be  the  area  of  cross  section  of  the  ring 
and  N'  the  total  number  of  turns, 

N'AB 


R  being  the  total  (ohmic)  resistance  of  the  secondary  circuit. 
(The  factor  io8  is  not  necessary  if  Q  and  R  are  in  absolute 
units.  It  must  be  used  when  Q  is  in  coulombs  and  R  in 
ohms  (p.  162).  If  the  throw  of  the  galvanometer  is  D 

Q=K-D 

when  K  is  the  ballistic  constant  (Exp.  LXIV).  If  K  is 
not  known,  a  calibrating  coil  for  determining  it  should  be 
included  in  the  arrangement  of  the  apparatus.  From  the 
above  formulae  and  the  data,  H,  B  and  fi  can  be  calculated. 


VVVVVVVS.'Vv'V'S 


FIG.  70. 

In  making  the  connection  for  the  practice  of  the  method, 
it  is  much  better  to  have  a  clear  understanding  of  the  plan 
and  purpose  of  each  part  and  to  proceed  systematically 
than  to  copy  the  connection  from  a  diagram.  In  the  first 


MAGNETIC    PERMEABILITY.  213 

place,  the  secondaries  of  both  coil  and  ring  should  be  kept 
permanently  in  series  with  the  galvanometer.  Then  a  switch 
is  to  be  so  arranged  that  the  current  can  be  passed  through 
either  the  primary  of  the  calibrating  coil  or  that  of  the  ring. 
A  suitable  rheostat  and  ammeter  are  needed  in  the  primary 
circuit.  If,  as  the  primary  current  is  increased,  the  deflec- 
tions of  the  galvanometer  become  too  great  to  be  read,  a 
resistance  must  be  put  in  series  or  in  parallel  with  the  gal- 
vanometer. The  former  is  preferable.  In  choosing  this 
added  series  resistance,  it  is  well  to  so  choose  it  that  the 
whole  new  secondary  resistance  is  made  a  simple  multiple 
of  the  former  resistance.  If  this  is  done  the  throw  will  be 
reduced  in  the  proportion  in  which  the  resistance  is  increased, 
and  all  throws  may  be  reduced  to  what  they  would  have  been 
with  the  original  resistance  by  multiplying  the  actual  throw 
by  the  proportion  in  which  the  secondary  resistance  was 
increased.  For  methods  of  bringing  the  galvanometer  to 
rest,  see  p.  156. 

Before  readings  are  begun  the  ring  should  be  demagnetized 
as  thoroughly  as  possible.  This  can  be  done  by  passing  an 
alternating  current  through  the  primary  and  reducing  it 
from  a  large  value  to  zero  by  means  of  a  rheostat,  or,  by 
rapidly  commutating  and  at  the  same  time  reducing  a 
direct  current.  Also  at  each  new  value  of  the  magnetizing 
current,  before  readings  are  taken,  the  commutator  should  be 
reversed  several  times,  so  that  the  iron  may  come  to  a  steady 
cyclical  state.  Instead  of  attempting  to  get  the  throw  on 
making  the  primary  current,  the  double  throw  on  reversing 
the  current  is  taken  with  both  calibrating  coil  and  ring 
and  divided  by  2. 

At  least  three  throws  that  agree  well  should  be  read  for 
each  strength  of  the  primary  current.  The  magnetizing 
current  should  be  increased  at  first  by  small  steps  to  bring 
out  the  characteristic  features  of  the  curve  of  magnetization, 
afterward  by  larger  steps.  The  work  need  not  be  continued 
after  the  readings  begin  to  differ  in  a  much  smaller  propor- 
tion than  the  successive  magnetizing  currents,  for  this  shows 


214  ELECTRICITY    AND    MAGNETISM. 

approaching  saturation.  The  throw  at  break  of  current 
should  also  be  carefully  noted  as  a  means  of  estimating  the 
permanent  magnetism;  for  from  the  throw  at  break  the 
diminution  of  B,  and,  therefore,  the  residual  value  of  B, 
can  be  calculated  as  above. 

In  the  report  the  various  values  of  /,  H,  Q,  B,  and  /* 
should  be  tabulated  and  a  curve  drawn  with  B  as  ordinates 
and  H  as  abscissae  (B-H  curve  or  curve  of  magnetization). 
On  the  same  sheet  a  B-fi  curve  should  also  be  drawn  and  a 
third  curve  showing  the  permanent  magnetism  as  deduced 
from  the  throws  at  break  of  the  current. 

Questions. 

1.  Why  is  only  the  ohmic  and  not  the  self -inductive  resistance  of 
the  secondary  considered  ? 

2.  What  is  the  effect  of  the  windings  being  closer  together  on  the 
inside  of  the  ring  than  on  the  outside? 

3.  What  is  meant  by  intensity  of  magnetization?     Susceptibility? 
Calculate  a  few  values  from  your  results. 


LXVI.  MAGNETIC  HYSTERESIS. 

Text-book  of  Physics  (Duff),  pp.  503-504;  Watson's  Physics,  pp.  716— 
722;  Hartley's  Magnetism  and  Electricity,  pp.  393-395.  Watson's 
Practical  Physics,  p.  561;  Henderson's  Electricity  and  Mag- 
netism, p.  294;  Ewing's  Magnetism  in  Iron,  Chapter  V. 

Let  a  magnetizing  force  applied  to  a  specimen  of  iron  as  in 
the  preceding  experiment  be  increased  step  by  step  and  let 
the  resulting  increases  of  magnetization  be  observed.  At 
some  stage  let  the  process  be  stopped  and  then  the  mag- 
netizing force  decreased  by  the  same  steps.  It  will  be  found 
that  the  steps  of  decrease  of  magnetization  are  less  than 
those  by  which  it  at  first  increased,  or  the  magnetization 
lags  behind  the  magnetizing  force.  This  is  called  hysteresis. 
For  a  complete  view  of  the  process  a  cycle  must  be  com- 
pleted, i.  e.,  the  magnetizing  force  must  be  decreased  step 
by  step  to  zero,  then  increased  to  a  negative  value  equal 
(numerically)  to  the  positive  value  at  which  the  decreases 
were  begun,  then  decreased  again  to  zero,  and  finally 


MAGNETIC    HYSTERESIS. 


215 


increased  again  to  the  highest  positive  value.  Thus  a 
hysteresis  loop  will  be  obtained. 

With  a  ring  specimen,  over  which  primary  and  secondary 
coils  are  wound,  there  are  two  methods  of  procedure. 

(A)  Step  by  Step  Method. — This  method  follows  closely 
the  general  description  given  above.  The  successive  steps 
are  indicated  in  figure  71.  The  increases  or  decreases  of 
7  must  be  made  without  break  of  the  current.  The  steps 
must  not  be  too  large  or  the  points  on  the  curve  will  be 
too  far  apart,  and  they  must  not  be  too  small  or  the  work 


-I   -I, 


FIG.  71. 

will  become  tedious.  To  satisfy  these  conditions,  place  in 
the  primary  circuit  a  special  rheostat  consisting  of  suitable 
resistances  in  parallel,  each  of  which  can  be  short-circuited 
by  a  knife-edge  switch.  Such  a  rheostat  may  be  made  up 
with  resistances  permanently  connected  in  position,  but  a 
better  plan  is  to  use  removable  resistances.  In  the  latter 
case  a  considerable  collection  of  units  should  be  supplied, 
and  from  these,  by  a  preliminary  trial,  units  that  will 
produce  suitable  changes  of  I  (e.  g. ,  from  4  to  o .  5  amp.  by  steps 
of  o.  5  amp.)  should  be  chosen  and  placed  in  position  in  the 
rheostat. 

It   is   not   necessary   to   start   from   zero   magnetization. 


2l6  ELECTRICITY   AND    MAGNETISM. 

Beginning  with  the  highest  current  to  be  used,  reverse 
several  times  to  produce  a  cyclical  state  and  then  find  the 
throw  on  reversal.  From  this  the  maximum  value  of  B  can 
be  calculated  as  in  Exp.  LXV.  Then  dimmish  the  current 
by  steps  and  note  the  throw  in  each  step.  After  the  step 
that  reduces  the  current  to  zero,  the  current  must  be 
reversed  and  the  resistances  increased  step  by  step.  The 
rest  of  the  process  needs  not  be  described.  From  each  throw 
the  corresponding  change  of  induction,  AJ5,  is  calculated 
as  in  Exp.  LXV.  When  the  cycle  has  been  completed  the 
algebraic  sum  of  the  throws  should  be  zero.  It  should  not 
be  necessary  to  change  the  sensitiveness  of  the  galvanometer; 
it  will  give  the  smaller  throws  with  less  accuracy,  but  they 
are  less  important.  This  "step  by  step"  method  of  meas- 
uring hysteresis  is  the  most  instructive  and  is  not  difficult 
after  some  initial  practice.  It  has,  however,  the  disad- 
vantage that  an  error  in  one  reading  of  the  galvanometer 
vitiates  the  whole. 

(B)  The  Ewing-Classen  Method. — The  last-mentioned  dis- 
advantage is  avoided  in  the  method  by  starting  each 
step  from  the  maximum  value  of  B.  As  before,  we  first  find 
by  reversals  the  value  of  B  corresponding  to  the  maximum 
value  of  /.  We  then  diminish  I  (without  breaking  the  cur- 
rent) and  from  the  throw  we  calculate  the  diminution  of 
B.  This  gives  us  a  second  point  on  the  curve.  We  then 
return  to  the  maximum  current  and,  after  several  reversals, 
to  re-establish  the  cyclical  state,  we  again  decrease  /,  but  by 
a  larger  amount  than  before.  From  the  throw  we  again 
calculate  the  diminution  of  B  and  thus  get  another  point 
on  the  curve.  Proceeding  in  this  way,  we  reach  the  stage 
at  which  I  is  decreased  from  its  maximum  to  zero.  This 
gives  us  the  point  at  which  the  curve  crosses  the  axis  on 
which  B  is  plotted. 

A  simple  method  of  producing  the  above  changes  of  7  is  to 
connect  the  rheostat  described  under  (A)  in  parallel  with  one 
of  the  cross-bars  of  the  Pohl's  commutator  used  for  reversing 
/  (Fig.  72).  If  this  cross-bar  be  suddenly  removed,  the 


MAGNETIC    HYSTERESIS. 


217 


resistance  in  the  rheostat  will  be  thrown  into  the  circuit 
without  breaking  the  current. 

By  the  above  process,  we  have  obtained  that  part  of  the 
descending  branch  of  the  hysteresis  loop,  which  lies  to  the 
right  of  the  B  axis.  To  obtain  the  remainder  of  the  branch, 
we  again  proceed  by  steps  from  the  positive  maximum  value 
of  B,  but,  since  each  change  of  /  will  carry  it  from  its  positive 
maximum  to  a  smaller  negative  value,  we  must  simultane- 
ously diminish  and  reverse  the  current.  To  be  able  to  do  this, 
remove  the  cross-bar  of  the  commutator  which  is  in  parallel 
with  the  rheostat  and  turn  the  commutator  so  that  the 


r 

FIG.  73. 


current  flows  to  the  ring,  but  does  not  pass  through  the 
rheostat  (Fig  73).  If  the  commutator  be  now  reversed,  the 
current  will  be  reversed  and  will  be  diminished  by  passing 
through  the  rheostat.  Thus  we  get  another  point  on  the  curve 
and,  by  a  series  of  such  steps  with  decreasing  resistances  in 
the  rheostat,  the  descending  branch  of  the  loop  is  completed. 
To  trace  the  other  branch  we  might  proceed  as  above,  begin- 
ning each  step  from  the  negative  maximum  of  B.  This, 
however,  is  unnecessary,  since  we  would  evidently  be  merely 
repeating  the  previous  readings.  The  loop  is  symmetrical 
about  the  origin,  and  the  co-ordinates  of  the  ascending  branch 


2l8  ELECTRICITY   AND    MAGNETISM. 

are  equal  to  those  of  the  descending  branch  but  with  signs 
reversed. 

It  can  be  shown  that  the  energy  expended  in  such  a  cyclical 
change  of  magnetization  is 


C 

xJ 


—     fidB 

4xJ 

ergs  per  c.c.  of  the  iron.  The  integral  also  represents  the 
area  of  the  loop,  due  allowance  being  made  for  the  scale  on 
which  it  is  plotted.  Hence  if  the  area  be  found  by  means 
of  a  planimeter  (the  use  of  which  will  be  explained  by  an 
instructor),  the  energy  loss  per  c.c.  per  cycle  can  be  cal- 
culated. 

The  total  number  of  lines  of  induction  through  each  turn  of  the 
magnetizing  coil  is  AB.  Since  the  total  number  of  turns  is  IN  , 
when  B  is  being  increased  there  is  induced  in  the  magnetizing  coil  an 
e.  m.  f. 

_,        d(lNAB}  jlB  r  n  c        ., 

E  =  --  -—  JT  —  -  =  —  VN-jT  C.  G.  S.  units. 
at  at 

V  being  the  volume  of  the  core  (  =  IA).  The  work  done  by  the 
battery  in  time  dt  in  overcoming  this  opposing  e.  m.  f.  is 

dW=IEdt=INVdB  ergs 
Now  the  area  of  the  hysteresis  loop  is  the  integral  of  HdB  and 


,W  -—   I  HdB 


Questions. 

1.  What  rise  of  temperature  would  1000  cycles  produce  in  the  iron 
if  no  heat  were  lost  ? 

2.  How  much   less   would   the   energy   loss   be  if  the   maximum 
magnetization  were  half  as  great  as  in  your  cycle? 


THE    MECHANICAL    EQUIVALENT    OF    HEAT.  2IQ 

LXVII.    (A)   THE  MECHANICAL  EQUIVALENT  OF  HEAT. 

(B)  THE  HORIZONTAL  INTENSITY  OF  THE 

EARTH'S  MAGNETISM. 

Text-book  of  Physics  (Duff),  pp.  557,  587-588;  Ames'  General  Physics, 
pp.  664-665,  688;  Watson's  Physics,  pp.  692-693,  674-677,  775- 
776;  Crew's  General  Physics,  §§  289,  319;  Hadley's  Electricity 
and  Magnetism,  pp.  336—340,  458;  Watson's  Practical  Physics, 
pp.  508-512. 

If  Q  calories  of  heat  be  produced  in  a  conductor  by  the 
passage  of  a  current  i  for  time  t,  and  if  no  other  work, 
chemical  or  mechanical,  be  performed,  then 

JQ=?Rtt 

]  being  the  mechanical  equivalent.  If  i  be  expressed  in  am- 
peres, R  in  ohms  and  Q  in  calories,  i2Rt  will  be  in  joules 
(one  joule  being  io7  ergs),  and  J  will  be  obtained  as  the 
number  of  joules  in  a  calorie. 

Q  can  be  measured  by  immersing  the  conductor  of  resistance 
R  in  a  known  mass  of  water  contained  in  a  vessel  of  known 
water  equivalent.  The  mass  of  water  may  be  obtained 
with  sufficient  accuracy  by  measuring  it  from  a  burette. 
To  eliminate  the  effects  of  radiation,  conduction  and  con- 
vection, the  water  should  be  at  the  beginning  of  the  passage 
of  the  current  as  much  below  the  temperature  of  the  room 
as  it  finally  rises  above  it,  for  the  current  is  kept  steady  and 
the  temperature  of  the  water  therefore  rises  steadily,  so 
that  it  is  as  long  above  the  room  temperature  as  below. 

The  resistance  R  may  be  measured  against  a  standard 
ohm  coil  by  Wheatstone's  Bridge,  and,  since  it  will  be 
found  necessary  to  use  a  wire  of  comparatively  small  re- 
sistance, R  should  be  measured  with  great  care.  Leads  of 
large  size  and  small  length  should  be  employed  for  con- 
necting the  wire  to  the  bridge.  While  being  measured  it 
should  be  immersed  in  the  calorimeter  in  water  at  the 
temperature  of  the  room,  so  that  the  mean  resistance 
throughout  the  experiment  is  obtained.  To  reduce  to 
absolute  units  the  resistance  in  ohms  is  multiplied  by  io9. 


220 


ELECTRICITY   AND    MAGNETISM. 


The  current,  i,  may  be  obtained  from  its  chemical  effect 
in  another  part  of  the  circuit.  Careful  measurements  have 
shown  that  unit  current  (C.  G.  S.)  flowing  through  a  solution 
of  copper  sulphate  of  a  certain  strength  between  copper 
electrodes  deposits  0.00326  gms.  of  copper  per  second  on 
the  cathode. 

The  form  of  copper  voltameter  employed  consists  of  a 
glass  vessel  containing  a  solution  of  copper  sulphate  into 
which  dip  three  plates.  The  two  outer  are  of  heavy  copper 


Calon'meter 


Rheostat 


Voltameter 
FIG.   74. 

and  are  both  joined,  directly  or  indirectly,  to  the  positive 
pole  of  the  battery,  forming  the  anode.  The  intermediate 
plate  is  thin  and  light,  and  is  connected  to  the  negative 
pole  of  the  battery,  forming  the  cathode.  A  satisfactory  solu- 
tion consists  of  15  grams  of  copper  sulphate  dissolved  in 
100  grams  of  water,  to  which  are  added  5  grams  of  sulphuric 
acid  and  5  grams  of  alcohol.  (The  alcohol  is  easily  oxidized, 
thus  preventing  the  oxidization  of  the  deposit  on  the 
cathode  and  the  formation  of  polarizing  compounds  at  the 
anode.) 


THE    MECHANICAL    EQUIVALENT    OF   HEAT.  221 

Clean  the  two  anode  plates  with  sand-paper  and  fasten 
them  in  the  two  outside  binding  posts  of  the  top  of  the 
voltameter.  Clean  with  sand-paper  a  cathode  plate,  wash 
with  tap  water  and  then  with  alcohol.  When  dry,  weigh 
on  one  of  the  chemical  balances,  weighing  to  milligrams 
with  the  rider.  Wrap  in  paper  and  set  aside.  Be  very 
careful  not  to  touch  with  the  fingers  any  part  of  the  plate 
which  will  be  in  the  solution,  after  it  has  been  cleaned. 
Clean  with  sand-paper  a  trial  cathode  and  mount  it  on  the 
middle  binding  post.  Before  putting  the  voltameter  in  the 
circuit,  dip  the  two  wires  which  are  to  be  connected  to  the 
voltameter  in  the  solution  of  the  voltameter.  Decrease  the 
variable  resistance  until  you  have  a  moderate  current,  but 
do  not  entirely  cut  it  out.  Notice  on  which  wire  copper  is  de- 
posited as  a  brown  powder.  Connect  this  wire  to  the  cathode 
of  the  voltmeter  and  the  other  wire  to  the  anode  plates. 

It  is  important  that  the  current  be  kept  constant.  It  is 
true  that  even  if  the  current  vary,  the  deposit  will  give  the 
true  mean  value  of  the  current.  But  what  is  needed  is  the 
mean  value  of  i2,  and  this  is  not  necessarily  the  same  as  the 
square  of  the  mean  value  of  i.  If  a  storage  battery  in  good 
condition  be  used  as  the  source  of  current,  the  current  will 
not  vary  much;  nevertheless,  a  tangent  galvanometer  or  an 
ammeter  should  be  included  in  the  circuit  to  test  the  con- 
stancy of  the  current.  There  is  also  another  reason  for 
including  a  current  meter  of  some  form.  The  difference 
of  potential  at  the  terminals  of  the  heating  coil,  or  iR, 
must  not  be  as  great  as  the  e.  m.  f.  (1.6  V)  that  will  elec- 
trolyze  water,  otherwise  some  part  of  the  energy  of  the 
current  will  be  spent  in  chemical  work.  Knowing  R,  one  can 
choose  a  safe  value  for  i.  If  the  constant  of  the  galvan- 
ometer be  not  known,  it  can  be  calculated  roughly  from 
the  dimensions  of  the  coils  and  the  approximate  value  (say 
o.i 8)  for  the  horizontal  component  of  the  earth's  field  (see 
Exp.  XLII),  and  so  the  deflection  corresponding  to  a  safe 
value  of  i  deduced.  An  ammeter,  if  available,  affords  a 
still  simpler  means. 


222  ELECTRICITY    AND    MAGNETISM. 

If  a  tangent  galvanometer  be  used  a  fairly  reliable  value 
for  the  horizontal  component'  of  the  earth's  field  may  be 
deduced  from  the  results  of  the  experiment.  For  this  pur- 
pose the  dimensions  of  the  galvanometer  should  be  care- 
fully measured  and  the  current  through  it  frequently  re- 
versed and  carefully  read.  The  Helmholtz  form  of  tangent 
galvanometer  may  be  used.  This  consists  of  two  coils  sepa- 
rated a  distance  equal  to  their  common  radius,  with  the 
needle  on  their  common  axis  midway  between  them.  This 
arrangement  of  two  coils  produces  a  very  uniform  field 
over  quite  an  area  where  the  needle  is  located,  allowing  the 
use  of  a  longer  needle.  The  formula  for  the  galvanometer 
(the  proof  for  which  will  be  found  in  text-books  on  physics)  is 

r    /       #2\f 
i  =H 1  i  H —  I    tan  ®. 


(If  a  simpler  type  of  tangent  galvanometer,  with  but 
one  coil,  is  used,  x  is  the  distance  from  the  plane  of  this  coil 
to  the  suspension  of  the  needle.) 

Equating  this  expression,  which  involves  the  dimensions 
of  the  galvanometer  and  the  deflection,  to  the  current  as  de- 
termined by  the  voltameter,  the  horizontal  component  is 
deduced. 

When  the  adjustments  have  been  completed,  open  the 
switch,  remove  the  trial  cathode  and  put  in  place  the  other 
cathode,  which  has  been  kept  wrapped  in  paper.  Take 
care  that  there  is  no  metallic  connection  between  the 
cathode  and  the  anode  plates.  Remove  all  iron  from  the 
neighborhood  of  the  tangent  galvanometer  and  from  your 
pockets.  All  wires  must  be  close  together  to  avoid  stray 
induction  and  the  galvanometer  had  best  be  at  some  distance 
from  the  other  apparatus. 

After  reading  the  temperature  of  the  calorimeter  every 
minute  for  five  minutes  note  the  exact  second  on  an  ordinary 
watch,  and  close  the  switch.  As  soon  as  possible,  read  both 
ends  of  the  needle.  Reverse  the  current,  making  the  reversal 
quickly,  and  again  read  both  ends  of  the  needle.  Always 


THERMOELECTRIC    CURRENTS.  223 

estimate  tenths.  Reading  both  ends  of  the  needle  eliminates 
error  due  to  the  axis  about  which  the  pointer  turns,  not 
coinciding  with  the  center  of  the  graduated  circle,  and 
reversing  eliminates  uncertainty  about  the  reading  for  the 
zero  position.  Keep  the  current  constant  with  the  variable 
resistance.  At  intervals  of  three  minutes  (approximately) 
read  both  ends  of  the  needle  and,  reversing,  again  read 
both  ends.  Read  the  temperature  every  minute  to  tenths 
of  the  smallest  division.  Allow  the  current  to  flow  until 
the  temperature  has  risen  to  the  extent  desired.  Note  the 
exact  second  of  breaking  the  circuit.  Continue  to  observe 
the  temperature  at  minute  intervals  for  five  minutes.  Re- 
move the  cathode,  being  very  careful  not  to  touch  the 
copper  deposit.  Wash  it  gently  with  tap  water  and  then 
with  alcohol,  allowing  the  liquid  to  simply  flow  over  the 
surface.  When  it  is  dry,  weigh  as  before.  Measure  very 
carefully  the  diameter  of  the  coils  in  a  number  of  directions, 
and  from  the  mean  determine  r.  Count  n,  the  total  number 
of  turns  in  both  coils,  and  measure  2  x,  the  distance  between 
the  centers  of  the  two  coils.  Weigh  the  inner  calorimeter 
vessel  and  note  of  what  metal  it  consists.  Plot  the 
temperature  readings  and  correct  for  radiation  (p.  63). 

Questions. 

1 .  Calculate  what  the  exact  voltage  at  the  terminals  of  the  heating 
coil  was. 

2.  What  sources  of  error  remain  uneliminated  ? 

3.  Calculate  the  mean  activity  of  the  current  in  the  coil  during  the 
experiment. 

4.  What    are    the    peculiarities    and    advantages    of    the  tangent 
galvanometer  ? 

5.  What  chemical  actions  take  place  in  the  voltameter?     To  what 
is  the  deposition  of  any  metal  proportional? 

6.  Why  is  it  advantageous  to  have  the  deflection  about  45°? 

LXVIII.  THERMOELECTRIC   CURRENTS. 

Text-book  of  Physics  (Duff),  pp.  593-598;  Ames'  General  Physics,  pp. 
679-683;  Watson's  Physics,  pp.  696-705;  Hartley's  Electricity 
and  Magnetism,  pp.  359-381. 

To  the  ends  of   wires  of   iron,  nickel,    silver    etc.,    cop- 
per wires  are  soldered  and   brought  to  binding   posts  on  a 


224  ELECTRICITY   AND    MAGNETISM. 

board.  Below  the  ends  of  the  board  are  vessels  contain- 
ing sand  or  oil  in  which  two  test-tubes  are  supported.  The 
junctions  are  placed  in  these  test-tubes  as  indicated  in 
figure  75.  The  binding  posts  are  connected,  by  copper 
wires,  to  a  key  of  as  many  parts  as  there  are  wires  to  be 
tested,  so  that  each  circuit  may  be  completed  through  a 
sensitive  galvanometer.  Thermometers  are  placed  in  the 
test-tubes  to  note  the  temperatures  as  one  vessel  is  being 
heated  by  a  burner. 

It  is  especially  important  that  the  temperature  should 
be  ascertained  accurately.     Hence   heat   should  be  applied 


Cu. 


Cu. 


FIG.  75. 

cautiously,  especially  at  first,  and,  when  observations  are 
to  be  made,  the  source  of  heat  should  be  removed,  and  time 
should  be  allowed  for  the  temperatures  to  become  fairly 
constant. 

The  galvanometer  reading  should  be  noted  with  the  great- 
est care  and  the  zero  should  be  frequently  tested.  After 
each  reading  of  the  galvanometer,  the  temperature  should  be 
noted. 

If  a  high  resistance  galvanometer  of  sufficient  sensitive- 
ness is  available,  the  other  resistances  may  be  neglected 
and  the  various  e.  m.  f.'s  will  then  be  proportional  to  the 
deflections.  Or  a  sensitive  low  resistance  galvanometer,  with 
a  constant  high  resistance  permanently  in  series,  may  be  used 
with  similar  simplicity.  The  constant  of  the  galvanometer, 
considered  as  a  voltmeter,  may  be  found  by  applying  to  it 
a  fraction  of  the  e.  m.  f.  of  a  standard  cell  (pp.  161,  179). 

With  this  arrangement  (which  will  be  readily  under- 
stood from  the  figure)  the  thermo-electric  force  of  each 
circuit,  consisting  of  copper  and  another*  wire,  may  be  de- 


ELEMENTARY    STUDY    OF   ALTERNATING    CURRENTS.       225 

termined.  Curves  representing  the  results  should  be  plot- 
ted with  the  differences  of  temperature  of  the  junctions  as 
abscissae  and  the  e.  m.  fs  as  ordinates. 

If  a  low  resistance  galvanometer  of  low  sensitiveness  is 
used,  it  will  be  necessary  to  consider  it  as  an  ammeter. 
In  this  case,  the  resistances  of  the  various  circuits  and  of 
the  galvanometer  must  be  found  and  the  constant  of  the 
galvanometer,  in  amperes  per  unit  deflection,  must  be  ob- 
tained by  connecting  it  in  series  with  a  standard  cell  and  a 
sufficient  known  resistance.  Thus,  the  currents  and  the 
resistances  being  known,  the  thermo-electric  forces  can  be 
calculated. 

Questions. 

1.  State  what   would   be  observed  if  the  temperature  of  the  hot 
junction  were  increased   steadily  beyond  the  highest  temperature 
used  in  this  experiment. 

2.  Is  the  effect  observed  here  due  solely  to  differences  of  potential 
produced  at  the  contacts  ? 


LXIX.  ELEMENTARY   STUDY   OF    RESISTANCE,    SELF- 
INDUCTION  AND  CAPACITY. 

Text-book  of  Physics  (Duff},  pp.  623—624;   Watson's  Physics,  p.  758; 
Hadley's  Electricity  and  Magnetism,  pp.  441—450. 

In  the  following  exercises,  which  are  intended  for  students 
who  have  not  made  a  study  of  the  theory  of  alternating 
currents,  some  of  the  properties  of  such  currents  are  studied 
and  compared  with  those  of  direct  currents. 

Ohm's  Law  for  steady  currents  states  that 

E 

—  =a  constant,  R, 

i 

where  R  is  called  the  resistance  of  the  conductor. 

(A)  Apply  various  e.  m.  f.'s  to  anon-inductive  conductor. 
Measure  the  current  by  an  ammeter,  and  the  voltage  by  a 
voltmeter  (of  any  type)  and  calculate  R  for  each  value  of 
the  e.  m.  f.  The  latter  may  be  varied  by  means  of  a  series 
rheostat. 


226  ELECTRICITY   AND    MAGNETISM. 

(B)  Apply  the  same  method  to  (i)  a  large  coil,   (2)  the 
large  coil  and  the  non-inductive  resistance,  (a)  in  series,  and, 
(b)  in  parallel.     Compare  the  results  of  (a)  and  (b)  with  the 
calculated  values. 

(C)  Repeat  (A)  and  (B),  using  alternating  currents  and 
an  electrostatic  voltmeter.     Corresponding  to  Ohm's  Law  we 
have 


E  / 

-  =constant  =  V  R 
i 

where  the  constant  is  called  the  impedance  and  L  is  the 
coefficient  of  self-induction  and  n  the  frequency.  Find  the 
value  of  the  constant  for  different  e.  m.  f.'s  and  currents, 
and,  from  the  mean  and  the  values  of  R  and  n,  calculate  L. 

Contrast  the  results  in  series  and 
parallel    combinations    with    the 
N  values  calculated  by  treating  im- 

-  1       pedance  in  the  same  way  as  re- 

Jtheo  I  —  —  i       .  .,  . 

A  I   sistance  in  direct  currents. 

Tabulate  all  results  so  that  they 
may  be  readily  compared. 

(D)   When   an  alternating  cur- 
F  rent  is  applied  to  a  condenser,  it 

is    charged,    discharged,    charged 

oppositely  and  discharged  during  each  alternation.  Evi- 
dently the  total  quantity  that  traverses  the  leads  in  each 
unit  of  time  is  proportional  to  the  frequency  and  to  the 
product  of  the  capacity  and  the  voltage  (since  q  =  CV)  and 
it  can  be  shown  that  the  current  is  given  by 

i  =  2nnCV. 

Measure  i  for  various  values  of  V  and  calculate  C.  Do 
this  for  several  condensers  (i)  separately,  (2)  in  parallel, 
(3)  in  series.  Compare  the  results  of  (2)  and  (3)  with  the 
calculated  values. 

Questions. 

i.  What  is  meant  by  the  effective  value  of  an  alternating  current 
and  what  ratio  does  it  bear  to  the  maximum  value? 


INDUCTION    AND    CAPACITY,    ALTERNATING    CURRENTS.     227 

2.  How,  by  means  of  a  diagram,  would  you  find  the  impedance 
when  given  the  ohmic  resistance  R  and  the  inductance  L  ? 

3.  Supposing   the    alternating  e.  m.  f.  resolved    graphically   into 
two  parts,  one  to  overcome  the  ohmic  resistance  and  the  other  to 
overcome  the  inductance,  what  relation  between  the  phases  of  these 
two  parts  does  question  (2)  suggest? 


LXX.  SELF-INDUCTION,    MUTUAL    INDUCTION    AND 
CAPACITY,  ALTERNATING  CURRENTS. 

See  references  to  LXIX.  J  .  J.  Thomson's  Electricity  and  Magnetism, 
§§233,  244-245;  Jackson's  Alt.  Cur.,  pp.  90-91,  151-200;  Parr's 
Electrical  Eng.  Testing,  pp.  222—224,  228—231,  234—235.  For 
Electrostatic  Voltmeter,  see  Parr,  367-371. 

This  exercise,  which  is  somewhat  more  advanced  than  the 
preceding,  is  intended,  for  students  who  have  made  some 
study  of  the  theory  of  alternating  currents. 

Let  E  be  the  alternating  e.  m.  f.  in  a  circuit  of  resistance 
R,  capacity  C,  and  self-induction  L.     If  i  is  the  current, 
.  E 


*>-  i  /Co)2 

We  can  test  the  above  formula  by  calculation,  after 
measuring  i,  E,  R,  C,  and  L.  An  inductance  with  a  magnetic 
core  has  a  variable  value  of  L,  the  magnitude  of  which 
depends  on  the  strength  of  the  current.  Hence,  for  this 
experiment,  an  inductance  consisting  of  a  very  large  coil 
containing  no  iron  is  used. 

(A)  Measurement   of  C.  —  If,   in  the   general   formula,    L 
be  zero,  C  can  be  deduced  from  the  values  of  i,  E,  and  R, 
assuming  that  <o,  which  equals  2  it  times  the   frequency  n, 
is  known.     E,  the  e.  m.  f.  across  the  terminals  of  the  con- 
denser, is  measured  by  an  electrostatic  voltmeter,  i  by  an 
alternating   current   ammeter.     Initially   a   high   resistance 
of  large  current  capacity  must  be  included.     This  may  later 
be  cut  out.     A  fuse  of  lower  capacity  than  the  range  of  the 
ammeter  must  be  permanently  in  circuit. 

(B)  Measurement   of  L.  —  The   value   of   L  is   found   by 
observing  the  values  of  i  and  £  in  a  circuit  containing  the 
self-inductance  coil  and  then  applying  the  general  formula. 


228  ELECTRICITY   AND    MAGNETISM. 

Sufficient  additional  resistance  must  be  placed  in  the  cir- 
cuit, but  the  value  of  E  required  is  that  across  the  terminals 
of  the  inductance  coil.  R,  which  in  this  case  is  the  resistance 
of  the  coil,  is  best  found  by  Wheatstone's  Bridge. 

(C)  Test    of   General    Formula. — Connect    the    condenser 
and  self-induction  in  series.     Measure  the  current  and  the 
total  e.  m.  f. ;  also  the  e.  m.  f.  across  each  part.     Connect  the 
condenser   and   inductance    coil   in   parallel.     Measure   the 
common  e.  m.  f.,  the  total  current  and  the  current  in  each 
branch. 

Calculate  i  in  the  series  arrangement  from  the  above 
formula  and  compare  with  the  experimental  value.  If  you 
are  familiar  with  the  method  of  complex  quantities  and 
graphical  methods,  apply  these  also  to  calculate  the 
currents  in  both  series  and  parallel  arrangements. 

(D)  Measurement    of  Mutual   Inductance. — Measure    the 
mutual  inductance,   M,  of  the  two  coils  of  a  transformer 
(with  iron  core)  by  observing  the  e.  m.  f.,  E,  across  one  coil 
when  a  measured  current,  i,  is  applied  to  the  other. 

E=Mi<o. 
Vary  i  several  times  and  find  how  M  varies. 

Questions. 

1 .  Why  is  an  electrostatic  voltmeter  necessary  ? 

2.  Does   the   self-induction   depend   upon    the   frequency?     Why 
does  the  latter  enter  into  the  equation  ? 

LXXI.  DIELECTRIC  CONSTANTS  OF  LIQUIDS. 

Text-book  of  Physics  (Duff),  pp.  533,  535;  Ames'  General  Physics,  pp. 
641,  66 1 ;  Watson's  Physics,  p.  637;  Hartley's  Electricity  and 
Magnetism,  Chapter  X. 

The  dielectric  constant  of  a  liquid,  or  the  ratio  of  the 
capacity  of  a  condenser  with  that  liquid  as  dielectric  to  its 
capacity  when  its  dielectric  is  air,  can  be  determined  by 
a  comparison  of  capacities  by  the  Bridge  Method  of  Exp. 
LX.  For  this  purpose  it  is  convenient  to  use  a  condenser 
consisting  of  two  parallel  plates,  the  distance  between  which 


DIELECTRIC    CONSTANTS    OF    LIQUIDS. 


229 


is  adjustable,  as  shown  in  figure  7  7 .  The  distance  between 
the  plates  can  be  measured  by  means  of  a  scale,  B,  attached 
to  the  movable  plate  A,  and  a  vernier  attached  to  the 
framework.  The  plates  hang  in  a  vessel  for  holding  the 
dielectric.  Two  methods  can  be  used.  In  one  the  distance 
between  the  plates  is  not  varied;  in  the  other  it  is  varied. 


A   A' 


B' 


p' 


FIG.  77. 


The  first  method  consists  in  comparing  the  capacity  of  the 
above  condenser  with  that  of  a  Ley  den  jar  (i)  when  the 
dielectric  is  the  liquid  to  be  tested;  (2)  when  it  is  air.  From 
these  comparisons  the  ratio  of  the  capacities  of  the  condenser 
in  the  two  conditions  is  deduced  and  this  equals  the  dielectric 
constant  of  the  liquid.  Instead  of  a  battery  and  galvan- 
ometer, an  induction  coil  and  a  sensitive  telephone  are  used. 

The  second  method  assumes  the  (approximate)  formula 


Ae 


C 


for  the  capacity  of  such  a  plate  condenser  (in  electrostatic 
units,  see  p.  162),  where  A  is  the  area  of  each  plate,  e  is  the 
dielectric  constant  of  the  surrounding  medium,  and  d  is 


230  ELECTRICITY   AND    MAGNETISM. 

their  distance  apart.    If  C  be  the  same  with  two  dielectrics, 
but  with  different  values  of  d 


e2     d2 

Having  obtained  a  balance  for  air  as  dielectric,  leave 
the  resistances  in  the  bridge  unchanged  and  again  obtain 
a  balance,  after  filling  the  jar  with  the  liquid  to  be  tested, 
by  adjusting  the  distance  between  the  plates.  This  second 
method  is  less  accurate,  since  the  formula  assumed  is  only 
approximate  and  the  distances  cannot  be  determined  as 
accurately  as  the  resistances.  An  accurate  formula  will  be 
found  in  Kohlrausch's  Physical  Measurements,  p.  379. 

The  above  methods  should  be  applied  to  two  highly 
insulating  liquids,  such  as  kerosene  and  benzol. 

Questions. 

1.  Calculate  the  capacity  of  the  two  plates  when  separated  by 
(a)  air;  (b)  liquid,  the  distance  apart  being  the  same  as  in  the  first 
part  of  the  experiment. 

2.  Calculate  the  charge  for  each  case  if  (a)  100  electrostatic  units 
of  potential  are  applied;  (b)  100  volts  (p.  162). 


LXIII.  ELECTRIC   WAVES   ON   WIRES. 

Dielectric  Constants  of  Liquids. 

Ames'  General  Physics,  pp.  752—754;  Watson's  Physics,  pp.  856-858, 
870—871;  Text-book  of  Physics  (Duff},  pp.  635-639;  Hadley's 
Magnetism  and  Electricity,  pp.  541-548,  585-586;  J.  J.  Thom- 
son's Electricity  and  Magnetism,  §  243;  Kohlrausch's  Physical 
Measurements  (on  capacity  of  a  plate  condenser),  p.  379.  Drude, 
An.  der  Phys.,  Vol.  8,  p.  336. 

In  this  experiment  electric  waves  on  a  wire,  AD,  are 
excited  by  electric  oscillations  in  a  neighboring  circuit  or 
"exciter,"  E,  which  contains  an  inductance,  L,  and  a 
capacity,  C.  The  period  of  such  oscillations  is  T=2x\/LC. 
The  inductance  is  that  of  two  thick  semicircular  wires. 
The  ends  e  and  e'  of  these  wires  carry  small  spheres  and 
the  wires  are  so  bent  that  the  spheres  are  beneath  the 


ELECTRIC    WAVES    ON    WIRES. 


231 


surface  of  kerosene  in  a  small  cup  and  form  a  spark- 
The  length  of  this  spark-gap  can 
be  adjusted  by  means  of  a  mi- 
crometer screw  attached  to  one 
of  the  ebonite  posts,  H  H',  on 
which  the  semicircular  wires  are 
supported.  The  condenser  is  of 
the  variable  form  shown  in  figure 
77  and  is  connected  between  the 
other  two  ends  of  the  semicircular 
wires.  The  impulses  that  start 
the  oscillations  in  the  exciter  are 
produced  by  a  Tesla  coil,  the 
secondary  of  which  is  connected 
across  the  spark-gap  ee't  while  the 
primary  of  the  Tesla  coil  is  con- 
nected through  another  spark-gap, 
Z,  to  the  secondary  of  an  induc- 
tion coil  /. 

The  wire,  AD,  on  which  the 
waves  are  formed  is  bent  to  a 
U-form  and  lies  in  a  horizontal 
plane  above  the  plane  of  the 
exciter.  The  oscillations  in  the 
exciter  act  by  induction  on  the 
part,  A,  of  the  wire  and  produce 
waves  that  move  along  the  wire 
toward  D;  between  any  two  cor- 
responding points,  such  as  d  and 
d',  there  is  an  oscillating  differ- 
ence of  potential  and  the  trans- 
mission of  this  oscillation  consti- 
tutes the  wave-motion.  At  the 
free  end,  D,  these  waves  are 
reflected  and  interfere  with  the 
direct  waves.  If  the  wire  is  of 
proper  length,  this  interference 


gap. 


B 


B 


D 

FIG.  78. 


232  ELECTRICITY   AND    MAGNETISM. 

produces  stationary  waves;  that  is,  the  wire  "resonates" 
to  the  exciter.  If  the  wire  be  also  sufficiently  long,  there 
will  be  one  or  more  nodes  on  the  wire,  that  is,  places  of  no 
potential  difference,  with  intervening  antinodes,  or  places  of 
maximum  oscillating  potential  difference.  If  a  small  wire 
"bridge,"  B,  be  placed  across  the  wire  at  a  node  it  will  not 
interfere  with  the  stationary  waves ;  but  it  will  destroy  them 
if  it  is  placed  at  any  other  point.  (It  will  be  instructive  to 
compare  the  above  with  the  formation  of  stationary  sound 
waves  in  a  resonance  tube  such  as  that  of  Exp.  XXX,  the 
tuning-fork  being  the  exciter.)  When  the  bridge  has  been 
placed  at  a  node  the  part  of  the  wire  between  it  and  the  free 
end,  D,  could  be  changed  in  length  or  removed  without 
appreciably  diminishing  the  oscillations  between  A  and  B, 
(just  as  the  lower  part  of  a  violin  string  that  is  touched  by 
the  finger  does  not  interfere  with  the  vibrations  of  the  upper 
part).  The  part  A  of  the  wire  is  (approximately)  a  node, 
although  it  is  the  part  where  the  oscillations  are  excited. 
(Compare  with  this  the  fact  that  when  a  tuning-fork, 
connected  to  a  long  thread  as  in  Melde's  experiment,  throws 
the  latter  into  stationary  vibrations,  the  point  of  connection 
is  a  node). 

Various  means  have  been  used  for  detecting  such  station- 
ary vibrations.  The  simplest  is  a  "vacuum"  tube,  V, 
which  contains  helium  at  a  very  low  pressure.  An  alter- 
nating potential  difference  between  the  ends  of  such  a  tube 
will  cause  oscillating  discharges  accompanied  by  a  glow. 
If  placed  across  the  wire  when  there  are  no  stationary 
oscillations  the  tube  will  not  glow;  but,  if  stationary  oscil- 
lations exist,  it  will  glow  brightly  at  an  antinode,  less 
brightly  between  an  antinode  and  a  node,  not  at  all  at  a  node. 
By  the  aid  of  the  tube  the  bridge  can  be  adjusted  to  each 
node.  If  the  wire  be  long  enough  to  permit  of  more  than 
one  node,  twice  the  distance  between  two  adjacent  nodes 
will  equal  the  wave  length. 

Since  the  wire  is  in  resonance  with  the  exciter,  the  period 
of  oscillation  of  the  wire  equals  that  of  the  exciter.  The 


ELECTRIC    WAVES    ON    WIRES.  233 

latter,  and  therefore  the  former,  can  be  changed  by  changing 
C.     Hence 


The  electric  waves,  while  directed  by  the  wire,  are  really 
waves  of  oscillation  of  electric  force  in  the  medium  between 
the  two  branches  of  the  wire.  Such  waves  travel  with  a 
velocity  that  is  independent  of  the  wave  length,  and,  if 
A!  be  the  wave-length  when  the  period  is  Tlt  ^  that  when 
the  period  is  T2,  v  =  ).l/Tl=A2/T2.  Hence 


By  determining  the  wave-length  with  air  as  the  dielectric 
in  C  and  then  with  a  liquid  as  dielectric,  we  can  evidently 
find  the  dielectric  constant  of  the  liquid. 

In  the  practice  of  the  method  most  trouble  is  likely  to  be 
due  to  the  spark-gap,  ee'.  It  must  be  adjusted  until  the 
spark  occurs  under  the  kerosene.  To  avoid  danger  to  the 
tube  by  accidental  dropping,  it  may  be  attached  loosely  to 
the  wire  by  a  loop  of  thread.  Each  node  should  be  deter- 
mined several  times.  The  distance  between  the  plates  of  C 
should  be  varied  four  or  five  times  with  air  as  dielectric 
and  a  curve  drawn  with  d  as  abscissa  and  X  as  ordinate. 
From  this  curve  and  the  value  of  X  for  each  liquid,  the  value 
of  the  dielectric  constant  for  that  liquid  is  readily  deduced. 

For  a  complete  proof  of  the  formula  works  on  electricity  and 
magnetism  must  be  consulted  (e.  g.,  J.  J.  Thomson's  Electricity  and 
Magnetism,  §  243).  The  following  considerations  suggest  the 
formula.  When  an  alternating  e.  m.  f.,  E,  is  applied  to  a  circuit 
containing  capacity,  self-inductance,  and  ohmic  resistance  in  series, 


The  potentials  across  the  self-inductance  (including  the  resistance) 
and  the  capacity,  respectively,  are 

£L  =  /-\/r2  +  a>2L2  and  Ec  =  ~^ 


234  ELECTRICITY  AND    MAGNETISM. 

Evidently  the  potentials  across  the  parts  of  the  system  may  be  greater 
than  the  total  e.  m.  f.  which  is  the  resultant  obtained  by  geometrical 
(i.  e.,'  vector)  addition  of  the  parts.  This  constitutes  resonance. 
It  is  complete  when 


Substituting  for  w  its  value  2x/T,  we  get  T  =  2n'\/LC.  This,  then, 
is  the  period  of  the  applied  e.  m.  f.  when  resonance  results.  It  is 
also,  therefore,  the  period  of  the  free  natural  vibrations  of  the  system. 
The  approximate  formula  for  the  capacity  of  a  plate  condenser  is 
given  in  Exp.  LXII,  a  more  exact  one  in  Kohlrausch,  p.  379. 

Questions. 

1.  Assuming  the  velocity  of  the  waves  to  be  that  of  light,  calculate 
the  frequency  of  the  oscillations  for  one  value  of  ^. 

2.  From  this  and  the  approximate  formula  for  C  calculate  L. 

3.  If  a  sufficient  amount  of  liquid  were  available,  how  could  its 
dielectric  constant  be  found  by  immersing  the  wire  AD  in  it? 

4.  What  would  be  observed  if  AD  were  contained  in  a  vacuum 
tube? 


TABLES 


236 


TABLES. 


TABLE  I. 
Logarithms  of  Numbers  from  i  to  1000. 


No. 

o 

i 

2 

3     4 

567 

8 

9 

10 

oooo 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

ii 

0414 

°4C3 

0492 

Q531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1  106 

13 

"39 

"73 

I2O6 

1239 

1271 

J3°3 

J335 

1367 

*399 

143° 

14 

1461 

1492 

1523 

*553 

1584 

1614 

1644 

1673 

!7°3 

1732 

IS 

1761 

1790 

1818 

1847 

.1875 

1903 

I931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

233° 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

262  5 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3°54 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

356o 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

37" 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

43*4 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

44  56 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

47J3 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5io5 

5"9 

5132 

5M5 

5159 

5172 

33 

5i85 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

539i 

5403 

54i6 

5428 

35 

544i 

5453 

5465 

5478 

549° 

5502 

55*5 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

57°5 

5717 

5729 

574° 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

4i 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

657i 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

691  1 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7°33 

7042 

7050 

7°59 

7067 

51 

7076 

7084 

7°93 

7101 

7110 

7118  7126 

7135 

7J43 

7152 

52 

7160 

7168 

7177 

7185 

7J93 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

73J6 

54 

7324 

7332 

7340 

7348 

7356 

7364  ',  7372  7380 

7388 

7396 

No. 


I     5 


TABLES. 


237 


TABLE  I.— Continued. 
Logarithms  of  Numbers  from  i  to  1000. 


No. 

0 

1  1  *  \  3 

4 

5  |  6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

745i 

7459 

7466  7474 

56 

7482 

7490 

7497 

7505 

75*3 

7520 

7528 

7536 

7543  7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619  7627 

58 

7^34 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694  7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767  7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839  !  7846 

61 

7853 

7860 

7868 

7875 

7882 

7889  7896 

7903 

7910  !  7917 

62 

7924 

7931 

7938 

7945 

7952 

7959  7966 

7973 

7980  7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048  8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116  8122 

65  8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182  8189 

66  8195  8202 

8209 

8215 

8222 

8228  8235  8241 

8248  8254 

67  8261 

8267 

8274 

8280 

8287 

8293  8299 

8306 

8312  8319 

68  8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376  8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439  |  8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500  !  8506 

7i  8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561  ;  8567 

72  8573 

8579 

8585 

859i 

8597 

8603 

8609 

8615 

8621  8627 

73  1  8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681  ;  8686 

74  8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739  8745 

75 

875i 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76  j  8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9oi5 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063  9069 

9074 

9079 

81 

9085 

9090 

9096 

9101  9106 

9112 

9117  i  9122 

9128 

9*33 

82 

9138 

9M3 

9149 

9154  9159 

9165 

9170  9175 

9180 

9186 

f3 

9191 

9196 

9201 

9206  9212 

9217  9222  i  9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

93°4 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

935° 

9355 

9360 

9365 

9370 

9375  I  938o 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425  i  943°  9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469  i  9474 

9479  9484 

9489 

89 

9494 

9499 

9504 

9509 

95J3 

9518  !  9523 

9528  9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

957i 

9576  9581 

9586 

9i 

959° 

9595 

9600 

9605  i  9609 

9614 

9619 

9624  9628 

9633 

92 

9638 

9643  !  9647 

9652  9657 

9661  9666 

9671  1  9675 

9680 

93 

9685 

9689  i  9694 

9699  9703 

9708  !  9713 

9717 

9722  9727 

94 

973i 

9736 

974i 

9745 

975° 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

997s 

9983 

9987 

9991 

9996 

No. 

o 

z 

2 

3 

4  I  5 

6 

7 

8 

9 

238 


TABLES. 


TABLE  II. 
Natural  Sines  and  Cosines. 


Sine 


Cosine 


4,   0 

o.oooo 

90 

I.OOOO 

I 

0.0175 

175 

89 

0.9998 

02 

2 

0.0349 

1  74 

88 

0.9994 

04 

3 

0.0523 

174 

87 

0.9986 

08 

4 

0.0698 

175 

86 

0.9976 

10 

5 

0.0872 

174 

85 

0.9962 

14 

6 

0.1045 

84 

0.9945 

7 

0.1219 

174 

83 

0.9925 

20 

8 

0.1392 

173 

82 

0.9903 

22 

9 

0.1564 

172 

81 

0.9877 

26 

10 

0.1736 

172 

80 

0.9848 

29 

ii 

0.1908 

172 

79 

0.9816 

32 

12 

0.2079 

171 

78 

0.9781 

35 

13 

0.2250 

171 

77 

0.9744 

37 

14 

0.2419 

169 

76 

0.9703 

15 

0.2588 

169 

75 

0.9659 

44 

16 

0.2756 

1  68 

74 

0.9613 

46 

J7 

0.2924 

168 

73 

0.9563 

50 

18 

0.3090 

166 

72 

0.9511 

52 

19 

0.3256 

166 

71 

0.9455 

56 

20 

0.3420 

164 

70 

0-9397 

58 

21 

0.3584 

164 

69 

0.9336 

61 

22 

0.3746 

162 

68 

0.9272 

64 

23 

0.3907 

161 

67 

0.9205 

67 

24 

0.4067 

1  60 

66 

°-9I35 

70 

25 

0.4226 

159 

65 

0.9063 

72 

26 

0.4384 

158 

64 

0.8988 

75 

27 

0.4540 

156 

63 

0.8910 

78 

28 

0.4695 

155 

62 

0.8829 

81 

29 

0.4848 

153 

61 

0.8746 

83 

30 

0.5000 

I52 

60 

0.8660 

86 

31 

0.5150 

I5° 

59 

0.8572 

88 

32 

0.5299 

149 

58 

0.8480 

92 

33 

0.5446 

147 

57 

0.8387 

93 

34 

0.5592 

146 

56 

0.8290 

97 

35 

0-5736 

144 

55 

0.8192 

98 

36 

0.5878 

142 

54 

0.8090 

102 

37 

0.6018 

140 

53 

0.7986 

104 

38 

0.6157 

139 

52 

0.7880 

106 

39 

0.6293 

136 

0.7771 

109 

40 

0.6428 

X35 

5° 

0.7660 

in 

41 

9-6561 

J33 

49 

0-7547 

XI3 

42 

0.6691 

130 

48 

Q-7431 

116 

43 

0.6820 

129 

47 

0.7314 

117 

44 

0.6947 

127 

46 

0-7*93 

121 

45 

0.7071 

124 

45°  * 

0.7071 

122 

Ctfsine 


Sine 


TABLES. 


239 


TABLE  III. 
For  Reduction  of  Time  of  Oscillation  to  an  Infinitely  Small  Arc. 


4  4      64  4 

If  t  =  observed  time  and 
T  =  true  reduced  time 
T-t-kt. 


a 

fc 

a 

k 

a 

& 

0° 

o.ooooo 

7° 

o  .00023 

14° 

O.OOOQ3 

i 

ooo 

8 

030 

J5 

107 

2 

002 

9 

°39 

16 

122 

3 

004 

10 

048 

J7 

138 

4 

008 

1  1 

058 

18 

154 

5 

012 

12 

069 

*9 

172 

6 

Oi; 

!3 

080 

20° 

I9O 

7° 

023 

14° 

093 

TABLE  IV. 
Reduction  of  Barometer  Readings  to  o. 

(The  corrections  below  are  in  mm.  and  are  to  be  subtracted.     The 
uncorrected  height  is  in  cm.) 


Temp. 

Brass  Scale 

Glass  Scale 

72 

73 

74 

75 

76 

77 

78 

74 

75 

76 

77 

78 

15 

r-75 

1.77  1.81  1.83 

1.86 

1.88 

1.91 

1.92 

1.94 

1.97 

2.0O 

2.  02 

1  6 

1.87 

1.89  1.93 

1.96 

1.98 

2.01 

2.03 

2.05 
2.17 

2.07 

2.  2O 

2.10 

2.13 

2.16 
2   29 

i7 

1.98 

2.01 

2.05 

2.08 

2.IO 

2.13 

2.16 

2.23 

2.26 

18 

2.IO 

2-13 

2.17 

2.  2O 

2.23 

2.26  2.29 

2.30 
2-43 

2-33 

2.36 

2-39 

2-43 
2.56 

"9 

2.22 

2.25 

2.29 

2.32 

2-35 

2.38  2.41 

2.46 

2-49 

2.53 

20 

2-33 

2.37   2.41 

2-44 

2.47 

2-51 

2-54 

2.56 

-59 

2.62 

2.66 

2.69 

2  I 

2-45 

2.48  2.53 

2.56 
2.69 

2.6O 

2.63 
2.76 

2.67 

2.68 
2.81 

2.72 

2.76 
2.89 
3.02 

3.15 
3.28 

2.79 

2.83 
2.96 

22 

2-57 

2.60  2.65 

2.72 

2.79 

2.85 

2.92 

23 

2.68 

2.72 

2-77 

2.81 

2.84 

2  88 

2.92 

2.94 

2.98 

3-11 
3.23 

3.06 
3-J9 

3-Jo 
3-23 
3-36 

24 

2.80 

2.84  2.89 

2.93 

3.05 

2.97 

3-°9 

3.01 
3-!3 

3-05 
3-J7 

3.06 

3-J9 

i 

25 

2.92 

2.96 

3.01 

3-32 

240 


TABLES. 


TABLE  V. 

Density  and  Volume  of  One  Gram  of  Water  at  Different 
Temperatures. 


Temp. 

Density 

Vol.  of  i.  gr. 

Temp. 

Density 

Vol.  of  i  .  gr. 

0° 

0.999878 

I.OOOI22 

J 

21° 

0.998065 

.001939 

i 

Q-999933 

.000067 

22 

0.997849 

.0021  56 

2 

0.999972 

.000028 

23 

0.997623 

.002383 

3 

0.999993 

.000007 

24 

0.997386 

.002621 

4 

I.OOOOOO 

.000000 

25 

0.997140 

.002868 

5 

0.999992 

.000008 

30 

0-99577 

.00425 

6 

0.999969 

.000031 

35 

0.99417 

.00586 

7 

Q-999933 

.000067 

40 

0.99236 

.00770 

8 

0.999882 

.000118 

45 

0-99035 

.00974 

9 

0.999819 

.000181 

50 

0.98817 

.01197 

10 

0-999739 

.000261 

55 

0.98584 

.01436 

1  1 

o  999650 

.000350 

60 

0-98334 

.01694 

12 

0.999544 

.000456 

65 

0.98071 

.01967 

*3 

0.999430 

.000570 

7° 

0.97789 

.02261 

U 

0.999297 

.000703 

75 

0-97493 

.02570 

i5 

0.999154 

.000847 

80 

0.97190 

.02891 

16 

0.999004 

.000997 

85 

0.96876 

-03225 

17 

0.998839 

.001162 

90 

0.96549 

•03574 

18 

0.998663 

.001339 

95 

0.96208 

.03941 

19 

0.998475 

.001527 

IOO 

0.95856 

•04323 

20 

0.998272 

.001731 

TABLE  VI. 

Density  of  Gases  (o°,  76  cm.).1 

Hydrogen 00008987 

Oxygen 0014290 

Nitrogen .0012507 

Air    0012928 

Chlorine 003167 

Carbon  monoxide 0012504 

Carbon  dioxide 0019768 

Ethane 001341 

Ethylene 001252 

Steam  (at  100°) 00060315 

Largely  from  Guye,  J.  Ch,  Phys.,  1907,  p.  203. 


TABLES. 


2AI 


TABLE  VII. 
Density  (o°),  Specific  Heat  (o°),  and  Coefficient  of  Linear  Expansion 


Element 

Density 

Specific 
Heat 

Coef.  of  Lin.  Exp. 
Multiplied  by  io6 

\luminum 

2    60 

2  2 

231 

Antimony 

6.62 

•  O4Q 

^  j.  x 

Bismuth    

9.8 

.OT,  I 

Cadmium  

8.61 

•  O  ?  ? 

-2Q.7 

Carbon,  diamond  
Carbon,  graphite  
Carbon    gas  carbon 

3-52 
2.25 

i  .00 

.10 
•15 

3r.?8 
7.8 

5      A 

Cobalt 

8.8 

.  I  06 

124 

Copper  
Copper  sulphate  (crys  ) 

8.92 
S-S8 

.094 

16.8 

Gold  

IQ.3 

.032 

M-4 

Iron  

7.8 

.  I  I 

I  2.1 

Lead 

n.^6 

O2  O 

202 

Magnesium  . 

1.74. 

Mercury 

I  "?  .  ^06 

QT.T.T. 

181   (cub.  exp  ) 

Nickel  

.IO8 

12.8 

Phosphorus  yellow  . 

1.83 

.20 

Phosphorus,  red  

2.IQ 

•  I  7 

Phosphorus,  metallic. 

2.^4 

Platinum 

214. 

O  3  3 

90 

Potassium  chloride 

I    08 

Silver  

I  O    ^  3 

.0  ^6 

I  Q.  2 

Sodium  chloride 

2.1  C 

Sodium  sulphate.  .  .  . 

2.6  t? 

Tin  

7.3 

.0  ^6 

22.3 

Zinc  

7-2 

•  OQ4 

20.  2 

Zinc  sulphate  (anhy  ) 

3       A  Q 

•4V 

16 


242 


TABLES. 


TABLE  VIII. 

Density,  Specific  Heat,  and  Coefficient  of  Expansion  of 
Miscellaneous  Substances  (o°). 


Substance 

Density 

Specific 
heat 

Coef.  of  Lin.  Exp. 
(Xio6) 

Castor  oil   

.060 

Glass,  green  
Glass   crown  

2.6 

2  .  7 

.19 
.  19 

8.9 
8.8 

Glass  crystal 

2    0 

.  18 

7  .  7 

Glass  flint 

3.  I  ^—  3    O 

.  10 

7  •  3 

Hard  rubber  
Marble  
Paraffin  

I-I5 

2-75 
.89 

7-7 
11.7 

Quartz,  crystal  II  .... 
Quartz,  crystal  -1-  
Quartz    fused 

2-653 
2-653 

2     2O 

.19 

7.2 
13.2 

•  54 

Alcohol  (ethyl)  
Benzol  

.81 
.800 

•54 
.^8 

i  .0481 
I.I761 

Carbon  bisulphide.  .  .  . 
Chloroform 

1.293 
I     *\3 

.24 

2  3 

I.I41 
i   n1 

Ether  (ethyl)  
Glycerine 

•74 
i   26 

•S3 

.  s8 

i-Si1 

TABLE  IX. 
Average  Value  of  Elastic  Moduli. 


Shear  Modulus. 


Coefficient  of  cubical  expansion  Xio3. 


Young's  Modulus. 


Brass     

V7Xiou 

10.  4X10" 

Iron  

7.7X10" 

I9-6XI011 

Steel 

8.2X10" 

22      Xio11 

TABLES.  243 

TABLE  X. 

Surface  Tension  T  (15°),  Temperature  Coefficient  of  Surface 
Tension  c',  and  Angle  of  Contact  a. 


T 

c' 

« 

Ethyl  ether  

10 

—  —  .  i  j 

1  6° 

Ethyl  alcohol 

2  ^ 

—  087 

0° 

Benzol  

31 

—•13 

0° 

Water  

76 

—  '5 

small 

Mercury  

527 

--38 

i35° 

TABLE  XL 

Coefficient  of  Viscosity  (20°). * 

Water oioo 

Mercury o  1 59 

Acetic  acid 0122 

Methyl  alcohol 00591 

Ethyl  alcohol .0119 

Ethyl  ether 00234 

Benzol 00649 

1  "\Vinkelmann,  1908,  I,  2,  p.  1397. 


244 


TABLES. 


TABLE  XII. 
Specific  Heats  of  Gases.' 


Temp. 


sv 


Argon... 20°  .1205 

Helium 20°  1.25 

Mercury 27  5°~3  56°  .0246 

Hydrogen o°-2oo°  3.406 

Nitrogen -3o°-20o°  .244 

Oxygen o°-2oo°  .217 

Air o°-2oo°  .2375 

Chlorine i9°-343°  -1  J  5 

Iodine 2oo°--377°  -0336 

Bromine 85°-228°  .0555 

Water i3o0-25o°  .480 

Hydrogen   sulphide...  io°-2oo°  .245 

Carbon  dioxide i  1 00°  .217 

Ammonia j  2o°-2io°  .512 

Chloroform !  28°-n8°  .144 

Ethyl  alcohol    i  io°-22o°  .453 

Ether 7o°-225°  .480 

Benzol n6°-2i8°  .375 


1.66 

1.64 

1.66 

1.396 

1.405 

1.40 

1-405 

1.32 

1.29 

1.29 

1.287 

1.28 

1.28 


1.14 
1.07 
1.187 


1  Jiiptner,  Phys.  Chem.  I,  pp.  71-73- 


TABLES.                                                       245 

TABLE  XIII. 

Pressure  of  Saturated  Water  Vapor  (Regnault). 

(mm.) 

Temp. 

Pressure                           Temp. 

Pressure 

| 

Ice                   Water                  29° 

29.782 

30 

3I-548 

—  10 

1.999                  2.078                  31 

33-405 

32 

35  359 

8 

2-379                  2.456                   33 

37-4io 

34 

39  565 

6 

2.821                   2.890                   35 

41.827 

40 

54.906 

4 

3-334                  3-387                   45 

7J-391 

50 

91.982 

2 

3-925                  3-955                   55 

117.479 

60 

148.791 

65 

186.945 

O 

4.600 

70 

233-093 

+     I 

4.940 

75 

288.517 

2 

5-302 

80 

354-643 

3 

5.687 

85 

433-41 

4 

6.097 

90 

525-45 

5 

6-534 

91 

545-78 

6 

6.998 

92 

566.76 

7 

7.492 

93 

588.41 

8 

8.017 

94 

610.74 

9 

8-574 

95 

633-78 

10 

9.165 

96 

657-54 

ii 

9.792 

97 

682.03 

12 

10.457 

98 

707.26 

13 

11.062 

98-5 

720.15 

14 

11.906 

99-o 

733-91 

15 

12.699 

99-5 

746.50 

16 

!3-635 

IOO.O 

760.00 

17 

14.421 

100.5 

773-71 

18 

15-357 

IOI.O 

787-63 

19 

16.346 

IO2.O 

816.17 

20 

i7-39i 

IO4.O 

875.69 

21 

18.495 

105 

906.41 

22 

19.659 

no 

1075.4 

23 

20.888 

I2O 

i49!-3 

24 

22.184 

I30 

2030.3 

25 

23-550 

X5° 

358i-2 

26 

24.998 

17S 

6717 

27 

26.505 

200 

1  1690 

28 

28.101                          225                 19097 

246 


TABLES. 


TABLE  XIV. 
Boiling  Point  of  Water,  t,  at  Barometric  Pressure,  p,(ui  m.) . 


p> 

t. 

P> 

t. 

P- 

t. 

740 

99.26° 

750 

99-630 

760 

100.00° 

41 

.29 

5i 

.67 

61 

.04 

42 

•33 

52 

.70 

62 

.07 

43 

•37 

53 

•74 

63 

.1  1 

44 

.41 

54 

.78 

64 

•15 

45 

.44 

55 

.82 

65 

.18 

46 

.48 

56 

•85 

66 

.22 

47 

•52 

57 

.89 

67 

.26 

48 

•56 

58 

•93 

68 

.29 

49 

•59 

59 

.96 

69 

•33 

75° 

99-63° 

760 

100.00° 

770 

100.36° 

TABLES. 


247 


TABLE  XV. 
Wet  and  Dry  Bulb  Hygrometer. 

(Actual  vapor  pressures  (mm.)  for  different  temperatures  of  dry 
thermometer  and  various  differences  of  temperature  between  the  two 
thermometers. 

The  first  vertical  column  gives  the  temperature  of  the  dry-bulb 
thermometer.  The  first  horizontal  line  gives  the  difference  between 
the  two  thermometers.  Since  the  difference  is  zero  if  the  air  is  satu- 
rated, the  second  vertical  column  gives  the  saturated  vapor  pressure 
for  the  corresponding  temperatures  in  the  first  column.) 


t°c. 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

I  I 

o 

4.6 

3-7 

2-9 

2.1 

J-3 

i 

4-9 

4.0 

3-2 

2.4 

1.6 

0.8 

2 

5-3 

4-4 

3-4 

2.7 

1.9 

I.O 

3 

5-7 

4-7 

3-7 

2.8 

2.2 

1'3 

4 

6.1 

5-1 

4.1 

3-2 

2-4 

1.6 

0.8 

5 

6-5 

5-5 

4-5 

3-5 

2.6 

1.8 

I.O 

6 

7.0 

5-9 

4-9 

3-9 

2.9 

2.0 

i.i 

7 

7-5 

6.4 

5-3 

4-3 

3-3 

2-3 

1.4 

0.4 

8 

8.0 

6.9 

5-8 

4-7 

3-7 

2.7 

*-7 

0.8 

9 

8.6 

7-4 

6-3 

S-2 

4.1 

3-1 

2.1 

i.i 

O.2 

10 

9.2 

8.0 

6.8 

5-7 

4.6 

3-5 

2-5 

i-5 

o-5 

ii 

9.8 

8.6 

7-4 

6.2 

5-1 

4.0 

2.9 

1.9 

0.9 

12 

10.5 

9.2 

8.0 

6.8 

5-6 

4-5 

3-4 

2-3 

i-3 

*3 

II.  2 

9.8 

8.6 

7-3 

6.2 

5-o 

3-9 

2.8 

i-7 

14 

II.9 

10.6 

9.2 

8.0 

6.7 

56 

4-4 

3-3 

2.2 

i.i 

15 

12.7 

n-3 

9-9 

8.6 

7-4 

6.1 

S-o 

3-8 

2.7 

1.6 

o-5 

16 

J3-5 

12.  1 

10.7 

9-3 

8.0 

6.8 

5-5 

4-3 

3-2 

2.1 

I.O 

I7 

14.4 

13.0 

n-5 

IO.I 

8-7 

7-4 

6.2 

4-9 

3-7 

2.6 

!-5 

0.4 

18 

i5-4 

I3.8 

12.3 

10.9 

9-5 

8.1 

6.8 

5-5 

4-3 

3-1 

2.O 

0.9 

19 

16.4 

14.7 

13-2 

11.7 

10.3 

8.9 

7-5 

6.2 

4-9 

3-7 

2-5 

1.4 

20 

17.4 

15-7 

14-1 

12.6 

ii.  i 

9-7 

8-3 

6.9 

5-6 

4-3 

3-i 

1.9 

21 

18.5 

16.8 

i5-i 

13-5 

12.  0 

10.5 

9.0 

7-6  6.3 

5-° 

3-7 

2-5 

22 

19.7 

17.9 

16.2 

14.5 

I2.9 

11.4 

9-9 

8.4|  7.0 

5-7 

4-4 

3-i 

23 

20.9 

19.0 

J7-3 

15-6 

13-9 

12.3 

10.8 

9-2 

7-8 

6.4 

5-1 

3-8 

24 

22.2 

20.3 

18.4 

16.6 

14.9 

13-3 

11.7 

IO.I 

8-7 

7-2 

5-8 

4-5 

25 

23.6 

21.6 

19.7 

17.8 

16.0 

14-3 

12.7. 

II.  I 

9-5 

8.0 

6.6 

5-2 

26 

25.0 

22.9 

21.  0 

19.0 

17.2 

i5-4 

J3-7 

12.  1 

10.5 

8.9 

7-4 

6.0 

27 

26.5 

24.9 

22.3 

20.3 

18.4 

16.6 

14.8 

I3-1 

11.4 

9.8 

8-3 

6.8 

28 

28.1 

25-9 

23-7 

21.7 

19.7 

17.6 

16.0 

14.2 

12.5 

10.8 

9-2  :  7-7 

29 

29.8 

27-5 

25-3 

23.1 

21.  1 

19.1 

17.2 

15-3 

13.6  11.9 

IO.2 

8.6 

3° 

31.6 

29.2 

26.9  24.6 

22.5  20-5 

18.5 

16.6 

14.7 

13.0 

I  1.2 

9.6 

248 


TABLES. 


TABLE  XVI. 
Vapor  Pressure  of  Mercury  (mm.). 


Temp. 

Pres. 

Temp. 

Pres. 

o 

O.O2 

170 

8.091 

+  20 

O.O4 

1  80 

I  I.OOO 

40 

O.o8 

190 

14.84 

60 

0.16 

200 

19.90 

80 

o-35 

210 

26.35 

IOO 

0.746 

22O 

34  7° 

no 

1.073 

230 

45-35 

I2O 

*-534 

240 

58.82 

I30 

2-175 

250 

75-75 

140 

3-059 

260 

96.73 

ISO 

4.266 

270 

123.01 

I  60 

5.900 

280 

I55-I7 

TABLE  XVII. 

Melting  Point  of  Metals.     (Holborn  and  Day  and 
Waidner  and  Burgess.1) 

Tin  .  . 232 

Cadmium 321 


Lead 

Zinc 

Antimony 

Aluminum 

Silver 

Gold 

Copper  .... 


327 
419 

63 1 

657 
961 
1063 
1084 


Platinum 1 770 


Rev.,  1909,  p.  467;   Compt.  Rend.,  1909,  cxlviii,  p. 


1177. 


TABLES. 


249 


TABLE  XVIII. 
Wave  Lengths  in  Angstrom  Units  (io-8  cm.). 


Line 

Element 

Wave  Length 

Color 

C  Ha     .    .     .. 

Hydrogen. 

6  ^63  o  ?4 

Red 

Dr 

Sodium 

c8o6  i  ?  ^ 

Yellow 

D, 

Sodium 

5890  182 

Yellow 

F  H^  

Hydrogen 

4861.  <C27 

Blue 

G'  Hv 

Hydrogen 

4.  -i  4.0.  6^4. 

Violet 

H      T"             . 

Calcium 

^068  62  s 

Violet 

Helium 

7O6  ^.2 

Red 

Helium 

6678.1 

Red 

Helium 

c8?  ?.6 

Yellow 

Helium 

t;oi  <.7 

Green 

Helium 

402  i.o 

Blue 

Helium 

471  ^.2 

Blue 

Helium 

4-47  I.  ^ 

Violet 

Mercury 

623  2   O 

Red 

Mercury 

C7QO.7 

Yellow 

Mercury 

^760-6 

Yellow 

Mercury 

^460.7 

Green 

Mercury 

40  ^0.7 

Green-Blue 

Mercury 

4916.4 

Blue 

Mercury 

4358.3 

Blue 

Mercury 

4078.1 

Violet 

Mercury 

4046.8 

Violet 

K~ 

Potassium 

7699.3 

Red 

Ko 

Potassium 

=:8^2.2 

Yellow 

^p  

fr  ;  

L»a  

Li  R 

Potassium 
Lithium 
Lithium 

4047.4 
6708.2 
6103.8 

Violet 
Red 
Orange 

Cadmium 

6438.5 

Red 

Cadmium 

5085.8 

Green 

Cadmium 

4799.9 

Blue 

2sO 


TABLES. 


TABLE  XIX. 

Refractive  Indices. 

[Yellow  light,  (D  lines)  20°.] 

Glass,  light  crown  (density  =  2  .  50) 

Glass,  heavy  crown  (density  =  3  .00) 

Glass,    light    flint    (density  ==2  .87) 

Glass,   heavy  flint  (density  =4.22) 

Quartz,  crystal,  -*- ,  ord 

Quartz,  crystal,  -1- ,  ext 

Alcohol,  ethyl 

Benzol, 

Carbon  bisulphide 

Chloroform 

Ether,  ethyl 

Glycerine 

Water 

Air,  o°,  7  6  cm i 


.5280 
.5604 
.5410 
7102 
•5442 
•5533 
.3614 
.5014 
.6277 
.4490 
•356o 
.4729 

•3329 
.000293 


TABLE  XX. 
Specific  Rotatory  Power  (20°).     Yellow  Sodium  Light  (D).* 


Active  Substance 


Concentration 
(=c)  (gr.iniooc.c.; 


Cane-sugar  R 

/    3-28 

66.639  —  .02o8<7 

Invert  sugar  L  

\  10-86 

I  —  14. 

66.453  —  .000124(7 

2O  07      —   O4l£ 

Glucose    (dextrose).    R 
(crystallized)  
Fructose  (levulose)  L  .  . 
Milk-sugar,  R  

Tartaric  acid  R  ... 

0-100% 
0-40 

f  ".' 

47-73    +.oisX% 
—  100.3      -f.ioSc 

52-53 
15.06    —.131(7 

Quartz  R  or  L  

1  22-63 

i3.436-.ii9C 
21  70  (for  i  mm  thickness) 

Landolt  and  Bornstein. 


TABLES. 

TABLE  XXI. 
Photometric  Table. 


251 


•  -.        .       x1  ui  at, 

^<-xj-pcn  \>    s.  n 

VJbVJlllClilll/    U< 

(3oo-w)2' 

n 

0 

I 

2 

3 

A 

50 

0.0400 

O.42O 

0.0440 

0.0460 

0.0482 

60 

.0625 

-65I 

.0678 

.0706 

•°735 

70 

.0926 

.0961 

.0997 

•1034 

.  1072 

80 

.1322 

.1368 

.1414 

.1643 

.1512 

90 

•1837 

.1896 

•!957 

.2018 

.  2082 

IOO 

•  25OO 

.2576 

•2653 

•2734 

.2815 

I  IO 

•3352 

•3449 

•3549 

•3652 

•3756 

I2O 

•4445 

•457° 

.4698 

.4829 

.4964 

130 

.5848 

.  6009 

•6i73 

•6343 

.6516 

I4O 

.7656 

.7864 

.8078 

.8296 

.8521 

J5° 

i  .0000 

i  .027 

1-055 

1.083 

1.113 

1  60 

1.306 

1.342 

1-379 

i  .416 

!-454 

170 

i  .  700 

i-757 

i.  806 

1.856 

1.907 

1  80 

2.250 

2-313 

2-379 

2.446 

2.516 

190 

2.983 

3.070 

3  .160 

3-253 

3-35° 

200 

4.000 

4.122 

4-285 

4-380 

4.516 

210 

5-444 

5.621 

5-803 

5-994 

6.192 

220 

7-563 

7.826 

8.100 

8.387 

8.687 

230 

10.80 

I  I  .21 

11.64 

12.09 

I2-57 

240 

16.00 

16.68 

17.41 

18.17 

18.98 

250 

25.00 

n 

5 

6 

7 

8 

9 

5° 

0.0504 

0.0527 

0.0550 

0.0574 

0.0599 

60 

.0765 

.0796 

.0827 

.0859 

.0892 

70 

.mi 

.1151 

.  1  193 

•1235 

.1278 

80 

•i563 

.1615 

.1668 

•!723 

.1779 

90 

.2148 

.2215 

.2283 

•2354 

.2428 

IOO 

.2899 

.2985 

•3074 

.3164 

•3256 

no 

.3864 

•3974 

.4088 

.4204 

•4323 

120 

.5012 

•5244 

•5389 

•5538 

.5691 

I30 

.6694 

.6877 

.7064 

•7257 

•7454 

140 

.8752 

.8988 

.9231 

.9481 

•9737 

I5° 

1-143 

1.174 

i  .205 

1-238 

1.272 

1  60 

1.494 

i-535 

i-577 

i  .620 

i  .664 

170 

i  .960 

2  .OI4 

2  .071 

2.129 

2.188 

180 

2.588 

2.662 

2-739 

2.817 

2.889 

190 

3-449 

3-552 

3-658 

3.768 

3.882 

200 

4.656 

4.803 

4-954 

5.111 

5-275 

2IO 

6.398 

6.122 

6-835 

7.068 

7.310 

22O 

9.000 

9-327 

9.670 

10.03 

10.40 

230 

13.07 

13.60 

14-15 

14.74 

15-35 

24O 

19.84 

20.75 

21.72 

22.75 

23.84 

250 

252 


TABLES. 


TABLE  XXII. 
Specific  Resistance  at  o°  C.  and  Temperature  Coefficient. 


Specific 
Resistance 

Temperature 
Coefficient 

Bismuth 
Copper  ( 
Copper  ( 
German 
Iron.  .  . 

(hard)     

132.6X10-° 
i  .590X10-° 
1.622X10-° 
20.24X10-° 
10.43X10-° 
19.85X10-° 
94.07  X  io-° 
8.957X10-° 
1.521X10-° 
1.652X10-° 
9.565X10-° 

0.0054 
0.0043 

annealed)  

hard  drawn) 

silver  (4Cu  +  2  Ni  +  i  Zn) 

0.00027 
0.007 
0.0039 
0.00089 
0.0034 
0.00377 

Lead  (pi 
Mercury 
Platinun 
Silver  (a 
Silver  (h 
Tin  

ressed)  
i  

nnealed) 

ard  drawn)  

0.004 

TABLE  XXIII. 
Specific  Resistance  and  Temperature  Coefficient  of  Solutions  (18°).* 


Sp.  Res. 

!   Temp. 
Coef. 

Sp.  Res. 

Temp. 
Coef. 

wHCl 

3-3  2 

i   o  0165 

nNaCl 

I  -7       A    f 

O    O2  2  6 

o.inHCl  
o.oinHCl  

28.5 

271. 

o.mNaCl.... 
o.oiwNaCl  .  . 

108.1 
974. 

wHNO3. 
o.iwHN03  ... 
o.oiwHNO3  .  . 

n  iH2SO4 

3-23 
28.6 

272. 

c    o  ^ 

0.163 
o  .  o  1  64   1 

wKCl  .. 
o.inKCl  

o.oiwKCl  .  .  . 

nAgNO3 

10.18 

89.5 
817. 

14.  .  7  c 

0.0217 
o  .  02  1  6 

o.iniH2SO4  .. 
o.oiw$H2SO4  . 

wC2H4O2 

44-4 
325- 

758 

1 

o.  in  AgNO3  ..  . 
o.oiwAgNO3  . 

w^Pb(NO3)2 

105-7 
922  . 

21.  8 

o.mC2H4O2  .. 
o.oiwC2H4O2  . 

2170. 
6990. 

o.miPb(N03)2 
o.omiPb(NO3)2 

129.4 
967. 

nNaOH  . 
o.iwNaOH  .. 
o.oiwNaOH. 

6-25 

54-7 
500. 

0.019 

w^ZnSO4  
o.i«£ZnSO4.. 
o.omiZnSO4 

37-6 
217. 
1362. 

0.02  5 

wNH4OH  .. 
o.mNH4OH  . 
o.oi«NH4OH 

1125. 

3°3°- 
10420. 

niCuSO4  . 
o.iw  JCuSO4.. 
o.oiw  ^CuSO4. 

38.8 
223. 
1385- 

0.0225 

*  A  normal  solution  (designated  by  the  subscript  w),  contains  in  one  liter  a 
number  of  grams  equal  to  the  chemical  equivalent  (atomic  or  molecular  weight 
divided  by  the  valency).  A  solution  with  the  subscript  o.  in  has  one-tenth  this 
concentration,  etc.  For  exmaple,  o.  iwHCl  has  3 .65  grs.  of  HC1  (gas)  in  one 
trile  of  solution,  or  that  proportion. 


TABLES.  253 

TABLE  XXIV. 
Dielectric   Constants. 


I 

II 

Hydrocyanic  acid  
Water  .  . 

....     96 
80 

Ether  .  . 

4(T 

Xylol  
Benzol 

2.26 

2   2 

Methyl  alcohol  
Ethyl    alcohol  
Ammonia  (liquid)  
Acetone  
Sulphur    dioxide  
Pyridene 

::::  % 

....        22 
....        17 
14 
....         12 

Toluol 

2   2 

Petroleum  

2.O7 

INDEX, 


Aberration,  138 
Absorption,  electric,  196 
Acceleration  of  gravity,  36 
Air,  density  of,  33 

thermometer,  76 
Alloys,  melting-point,  109 
Alternating     current     measurements, 

225,  227 

Ammeter,  calibration  of,  198 
Anderson's  method    (self   induction), 

204 

Angle  of  prism,  131 
Angular  field  of  view,  143 
Apparent  expansion  of  gas,  76 

of  liquid,  74 
Arc  of  vibration,  correction,  239 

Balance,  21-25 

correction  for  air  buoyancy,  24 

method  of  oscillations,  22-24 

ratio  of  arms,  24 
Ballistic  galvanometer,  156,  208 
Barometer,  21 

table  of  corrections,  239 
Battery,  electromotive  force,  191,  193 

resistance,  185,  195 
Beckman  thermometer,  65 
Biquartz,  151 
Bismuth  spiral,  207 
Boiling-point  of  water  (table),  246 
Bridge,  Wheatstone's,  153 
Bunsen  photometer,  126 

Cadmium  cell,  160 
Calibrating  coil,  209,  212 
Calibration  of  ammeter,  198 

of  galvanometer,  179 

of  resistances,  183 

of  scale,  26 

of  thermometer,  67 

of  voltmeter,  196 

Callender's  equation  (platinum  ther- 
mometer), 117 


Calorimeter,  for  gases,  113 

for  liquids,  114 

for  solids,  no,  in 

simple,  89 

Candle-power,  measurement  of,  1 26 
Capacity,  absolute  measurement,  203 

divided  charge  method,  178 

measurement     (alternating     cur- 
rents), 225,  227 

Capacities,  comparison  of,   200,   228, 
230 

different  types,  194,  229 
Carey  Foster  bridge,  183 
Cathetometer,  19 
Chemical  hygrometer,  84 
Chromatic  aberration,  138 
Clark  cell,  160 
Clement    and    Desermes'    method 

(specific  heat  of  gases),  91 
Coefficient  of  apparent  expension,  74, 
76 

of  expansion,  71,  241,  242 

of  friction,  41-45 

of  increase  of  pressure,  76 

of  mutual  induction,  206 

of  self  induction,  203 

of  viscosity,  56,  243 
Coincidence  method,  38 
Commutator,  double,  161 
Comparator,  15 
Condenser,  see  capacity. 
Conductivity,  thermal,  102 

of  electrolyte,  188 
Copper  voltameter,  220 
Curves,  plotting  of,  1 1 

Daniell  cell,  159 
Demagnetization  of  iron,  213 
Density,  of  gases,  33,  240 

of  liquids,  29 

of  powders,  32 

of  solids,  28,  241,  242 

of  water,  240 


255 


256 


INDEX. 


Dew-point,  83 

Dielectric  constant,  228,  230 

(table),  253 

Diffraction  grating,  147 
Dip  circle,  167 
Dividing  engine,  ry 
Dolezalek  electrometer,  176 
Double  bridge,  178 
Double  commutator,  161 
Drude's    apparatus    (electric   waves), 
230 

Earth  inductor,  169 
Elastic  constants,  242 
Electric  absorption,  196 
Electrical  resonance,  232 

units,  162 

waves,  230 

Electrolytes,  resistance  of,  188 
Electrometer,  quadrant,  176 
Electromotive  force,  device  for  small, 
161 

measurement  of,  191,  193 

of  various  cells,  160 
Equivalent,  chemical,  31 
Errors,  2-10 

of  weights,  27 
Expansion,  apparent,  74,  76 

coefficient  of,  71,  241,  242 
Eutectic  alloy,  109 

Focal  length  of  lenses,  137,  140 

of  mirrors,  134 

Frequency  of  tuning  fork,  44,  1 20 
Friction,  coefficient  of  kinetic,  42 

coefficient  of  static,  41 

"G,"  determination  of,  36 
Galvanometer,  bringing  to  rest,  156 

calibration  of,  179,  208 

damping,  157 

different  types,  155 

resistance  of,  171,  173 

shunt,  157 

study  of  ballistic,  158,  208 

tangent,  222 

Gas,  coefficient  of  increase  of  pressure, 
76 

density  of,  240 
Grating,  diffraction,  147 

Heat,  conductivity  for,  102 
Heat  value  of  gas,  113 

of  liquid,  114 

of  solid,  no 


Hempel  calorimeter,  no 

Hooke's  law,  46 

Horizontal  component  of  earth's  field, 

163,  219 
Hygrometry,  83 
Hypsometer,  70 
Hysteresis,  214 

Incandescent  lamp,  study  of,  128 

Inclination,  magnetic,  167 

Index  of  refraction,  measurement  of, 

132 

table  of,  250 

Inertia,  measurement  of  moment  of,  5  2 
Insulation  resistance,  176 
Interferometer,  149 
Iron,  permeability  of,  210 

Junker  calorimeter,  113 

Kundt's  method  (velocity  of  sound)' 
122 


Latent  heat  of  fusion,  94 

of  vaporization,  96,  99 
Lenses,  combinations,  140 

focal  length,  137 

rule  of  signs,  125 
Light,  filters,  124 

monochromatic,  124 
Logarithmic  decrement,  157 

tables,  236 
Low  resistance,  measurement  of,  178- 

183 

Lummer-Brodhun     photometer,    126, 
127 

Magnetic  field,  measurement  of,  207 
of  earth,  dip,  167 
of  earth,  horizontal  component, 

163,  219 
hysteresis,  214 
permeability,  210 
Magnetometer,  165 
Magnification,  141 
Magnifying  power  of  telescope,  142 
Mance's  method  (battery  resistance), 

185 

Mechanical  equivalent  of  heat,  elec- 
trical method,  219 
by  friction,  105 
Melting-point  of  alloy,  109 
of  metals  (table),  248 


INDEX. 


257 


Mercury,   vapor  pressure  of   (table), 

248 

Michelson's  interferometer,  149 
Micrometer  caliper,  14 

microscope,  15 
Minimum  deviation,  132 
Mirror  and  scale,  adjustment  of,  25 
Mirrors,    spherical,    measurement    of 
focal  length,  134 

rule  of  signs,  125 
Moduli,  law  of,  31 
Mohr-Westphal  balance,  29 
Moment  of  inertia,  52 
Monochromatic  light,  124 
Mutual  induction,  206,  227 

Optical  lever,  48,  73 
pyrometer,  116 

Passages,  method  of,  54 

Pendulum,  correction  for  arc  (table), 

239 

physical,  37 
•  simple,  36 
Permeability,  210 
Photometric  table,  251 
Photometry,  126 
Pirani's  method   (mutual  induction), 

200 

Pitch  of  tuning  fork,  44,  120 
Planimeter,  218 
Platinum  thermometer,  116 
Pohl  commutator,  202,  216 
Polarization,  rotation  of  plane  of,  150 
Possible  error,  4-9 
Post-office  box  bridge,  154 
Potentiometer,  197 
Pressure,  coefficient  of  increase  of,  76 

of  mercury  vapor,  248 

of  water  vapor  (measurement),  80 

(table),  245 
Primus  burner,  114 
Prism,  angle  of,  131 

minimum  deviation,  132 
Probable  error,  9 
Pyknometer,  33 
Pyrometry,  115 

Quadrant  electrometer,  176 

Radiation  correction,  63 

pyrometer,  116 
Radius  of  curvature  of  mirror,  134 


Ratio  of  specific  heats,  measurement 
of,  91,  122,  123 

table,  244 
Refractive  index,  measurement  of,  132 

of  lenses,  136 

table,  250 
Regnault's  apparatus,  hygrometry,  83 

vapor  pressure,  81 
Reports,  2 
Resistance,  boxes,  153 

electrolytic,  188 

high,  175-178 

low,  178-183 

measurement  of,  169 

of  ballistic  galvanometer,  208 

of  battery,  185,  195 

of  galvanometer,  171,  173 

temperature  coefficient  of,  186 
Resistances,  comparison  of,  183 
Resolving  power,  of  eye,  146 

of  telescope,  145 
Rigidity  of  metals,  51 
Rosenhain  calorimeter,  in 
Rotation    of    plane    of    polarization, 
measurement  of,  150 

table,  250 
Rubber  grease,  34 

Saccharimetry,  150 
Scale,  calibration  of,  26 
construction  of,  26 
Self    induction,     alternating     current 

method,  225,  227 
Anderson's  method,  204 
inductions,  comparison  of,  205 
Shunts,  galvanometer,  157,  171 
Shear  modulus,  51 
Signs  (mirrors  and  lenses),  125 
Slide  wire  bridge,  154 
Sound  velocity  of,  119,  122 
Specific  gravity  bottle,  33 
heat,  of  gases,  91 
(table),  244 
of  metals,  85 
(table),  241 
of  miscellaneous  substances 

(table),  241,  242 
inductive  capacity,  see   dielectric 

constant, 
resistance  of  electrolytes,  188,  252 

of  metals,  171,  252 
rotatory  power  (table),  250 
Spectrometer,  130 
Spherical  aberration,  138 
Spherometer,  16 


258 


INDEX. 


Standard  cells,  159. 
Stroboscopic  disk,  44 
Surface  tension,  measurement  of,  60 
table,  243 

Tangent  galvanometer,  222 
Telescope,  adjustment  of,  25 

magnifying  power  of,  142 

resolving  power  of,  145 
Temperature  coefficient  of  expansion, 

7*>  74 

of  expansion  (tables),  240-242 

of  resistance,  186 

(table),  252 

Thermal  conductivity,  102 
Thermocouple,  91,  116,  223 
Thermometer,  air,  76 

Beckman,  65 

calibration  of,  67-7 1 

fixed  points,  69 

platinum,  116 
Thomson's  double  bridge,  181 

method  (galvanometer  resistance), 

method  of  mixtures,  202 
Time  of   vibration,  method    of   coin- 
cidences, 38 
method  of  passages,  54 
reduction  to  infinitely  small  arc, 

239 

signals,  25 

Torsion,  modulus  of,  51 
Trigonometrical  functions  (table),  238 
Tuning  fork,  pitch  of,  44,  1 20 

Units,  electrical,  162 


Vacuum,  reduction  of  weighing  to,  24 

Valson's  law  of  moduli,  31 

Vapor  pressure  of  mercury  (table),  248 

of  water  (measurement),  80 

(table),  245 

Velocity  of  sound,   Kundt's  method, 
122 

resonance  method,  119 
Vernier,  13 

caliper,  14 

Virtual  image,  136,  139 
Viscosity,  measurement  of  coefficient 
of,  56 

table,  243 

Voltameter,  copper,  220 
Voltmeter,  calibration,  196 
Volumenometer,  32 
Water,  boiling-point  (table),  246 

density  (table),  240 

equivalent,  89 

vapor  pressure,  80,  245 
Wave  length,  of  electric  waves,  230 

of  light  waves  (measurement),  147 
(table),  249 

of  sound  waves,  119,  122 
Weighing,  by  oscillations,  22 

double,  24 

reduction  to  vacuum,  24 
Weight  thermometer,  76 
Weights,  calibration  of,  27 
Weston  (cadmium)  cell,  160 
Wet  and  dry  bulb  hygrometer,  84,  247 
Wheatstone's  bridge,  153,  169 

Young's  modulus,  by  bending,  47 
by  stretching,  46 
table,  242 


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RECALL 


LIBRARY,  UNIVERSITY  OF  CALIFORNIA,  DAVIS 

Book  Slip-Series  458 


167U18 


Duff,  A.W. 

Physical  measure- 
ments . 


Call  Number: 

QC37 

D8 

1910 


QC37 


167418 


